On Sunday, August 29, 2021 at 6:35:52 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
> >
> > There is no "point next to zero",
> >
> Then there is a gap.

No, there is no gap.

Hint: ~Ex e IR: Ay e (0, 1]: 0 < x < y .

Actually: (0, 1] u {0} is a compact interval.

On 8/29/2021 12:31 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. August 2021 um 06:58:57 UTC+2:

>> In order for x to be next to 0,
>> there must be no point y between 0 and x,
>> there must be no point y in (0,x).
>> In other words, (0,x) must be a gap.
>
> You forget that dark points cannot be analyzed
> as individuals or ordered.

A gap with points in it is not a gap.
Please do not forget this.

> There are ℵo dark points in the interval (0, 1/n)
> where n is the largest known or defined natural number.
> This status is not permanent.

For each 1/n, a steppable 0,...,n exists in which,
for adjacent i,j, j = i+1.
This status is permanent.

>> Consider the unit fractions 1/k.
>> There is no open segment (0,1/k) in which
>> there are no unit fractions (see 1/(k+1)).
>> There is no (0,1/k) which is a gap.
>> No 1/k is next to 0.
>
> No definable 1/k is next to 0.
>
> But either there is something next to 0 or nothing.
>
> This are the alternatives:
> A defined point or not a defined point.

For each interval (0, d), there is a unit fraction 1/k
such that a steppable 0,...,k exists in which,
for adjacent i,j, j = i+1.

(Otherwise, the line does not have the Archimedean property.)

Therefore, each interval (0, d) is NOT a gap.

What is there that can be next to zero
without there being a gap between it and zero?
Nothing.

On Sunday, August 29, 2021 at 10:06:47 AM UTC-7, Jim Burns wrote:
> On 8/29/2021 12:08 PM, WM wrote:
> > William schrieb
> > am Samstag, 28. August 2021 um 22:01:23 UTC+2:
> >> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3,
> >> WM wrote:
>
> >>>> It is not true that there must be
> >>>> a gap next to zero, or a point next to zero.
> >>>
> >>> Either there is a definable point next to zero or
> >>> there is no definable point next to zero.
> >>> Do you agree?
> >>
> >> There is neither a point ("definable" or not)
> >> nor a gap next to zero.
> >
> > What is next to zero?
> > Note that "nothing" is synonymous to "gap".
> Note that "nothing" is NOT synonymous with "gap".
>
> To borrow a term from art, a gap is _negative space_
> It is an interval in which points are NOT.
> https://en.wikipedia.org/wiki/Negative_space
>
> In order for there to be a point next to zero,
> there must be negative space next to zero, and then the point.
> If there is no negative space, there is no next point.
> If there is no negative space, there is no gap.

Least Upper Bound is the usual axiom attached to
the ordered field to result the gapless complete ordered field.

(That otherwise "the least upper bound is defined as
being in the set except that least upper bound LUB holds".)

The correct word for continuous in the discrete sense is contiguous.

(That the definition of continuous function applies to
domains that preserve a metric.)

A usual beads-on-a-string or the course-of-passage through
the infinitely divided by motion or time, it's a linear continuum.

The complete ordered field of course is mostly "we can use
and develop this definition through about three, seven, or
ten years of instruction". Or more....

On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:

> What is next to zero? Note that "nothing" is synonymous to "gap".

Nope
A gap next to 0 is an interval [0,x] (note x must be a real number, not a "dark real number" whatever that is) that does not contain an element of U. There is no such interval so there is no gap. So there is neither a gap nor a point next to 0.

--
William Hughes

On Sunday, August 29, 2021 at 11:09:41 AM UTC-7, William wrote:
> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>
> > What is next to zero? Note that "nothing" is synonymous to "gap".
> Nope
> A gap next to 0 is an interval [0,x] (note x must be a real number, not a "dark real number" whatever that is) that does not contain an element of U. There is no such interval so there is no gap. So there is neither a gap nor a point next to 0.
>
> --
> William Hughes

Points in a point's neighborhood are in all the points' neighborhoods.

On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> Jim Burns wrote:

>> To borrow a term from art, a gap is _negative space_
>> It is an interval in which points are NOT.
>> https://en.wikipedia.org/wiki/Negative_space
>>
>> In order for there to be a point next to zero,
>> there must be negative space next to zero, and then the point.
>> If there is no negative space, there is no next point.
>> If there is no negative space, there is no gap.
>
> Least Upper Bound is the usual axiom attached to the
> ordered field to result the gapless complete ordered field.

Here, I'm using "gap" in a more restricted sense than I have
elsewhere. Call them "multi-point gaps".

The rationals are already sufficient for a multi-point-gapless
ordered field. No multi-point interval (a,b) with a ~= b
is without rationals. No (a,b) is a gap.

The Least Upper Bound property eliminates single-point gaps.

One important consequence of the elimination of single-point
gaps is that we can formally describe and reason about
Achilles chasing the tortoise along a _continuous_ path.
Well, we could use the same definitions _without_ LUB, but they
won't mean what we want them to mean. The path Achilles takes
can be continuous _at every point_ but, if there are
single-point gaps, he can execute arbitrarily large teleports
across that gap. Sure, that wouldn't be continuous _if there_
_was a point there_ but there isn't.

That is the reason (one, at least) that we need the LUB
property: so that "continuous at all the points"
works out to "no teleporting".

> (That the definition of continuous function applies to
> domains that preserve a metric.)

Continuous functions don't need to preserve a metric.

Rotations and translations preserve the metric.

Something along those lines that is true:
The inverse f^-1 of a continuous function f maps open sets
to open sets. We could say f^-1 preserves the topology.

On Sunday, August 29, 2021 at 9:35:52 AM UTC-7, WM wrote:
> Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
> > On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:
>
> > > There is nothing between the point next to zero and <bla>
> >
> > There is no "point next to zero",
> Then there is a gap. Then the real line is discontinuous. That is possible.

Note that on this criterion, the integer line is 'continuous', thus contradicting our illustrious Professor's idiosyncratic daffynition in his 'best-selling' 'textbook'

WM formulated on Sunday :
> Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
>> On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:
>
>>> There is nothing between the point next to zero and <bla>
>>
>> There is no "point next to zero",
>
> Then there is a gap. Then the real line is discontinuous. That is possible.

Show how a complete ordered field is discontinuous.

On Sunday, August 29, 2021 at 9:09:02 AM UTC-7, WM wrote:

> What is next to zero? Note that "nothing" is synonymous to "gap".

https://dyn1.heritagestatic.com/lf?set=path%5B1%2F0%2F9%2F5%2F1%2F10951614%5D%2Csizedata%5B850x600%5D&call=url%5Bfile%3Aproduct.chain%5D

On Friday, August 27, 2021 at 4:14:36 AM UTC-7, WM wrote:
> Jim Burns schrieb am Donnerstag, 26. August 2021 um 23:17:48 UTC+2:
> > If you can prove arithmetic is inconsistent, go ahead.
> Arithmetic is consistent.

Proof by Casting a Spell

On Sunday, August 29, 2021 at 11:24:51 AM UTC-7, Jim Burns wrote:
> On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> > On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> > Jim Burns wrote:
>
> >> To borrow a term from art, a gap is _negative space_
> >> It is an interval in which points are NOT.
> >> https://en.wikipedia.org/wiki/Negative_space
> >>
> >> In order for there to be a point next to zero,
> >> there must be negative space next to zero, and then the point.
> >> If there is no negative space, there is no next point.
> >> If there is no negative space, there is no gap.
> >
> > Least Upper Bound is the usual axiom attached to the
> > ordered field to result the gapless complete ordered field.
> Here, I'm using "gap" in a more restricted sense than I have
> elsewhere. Call them "multi-point gaps".
>
> The rationals are already sufficient for a multi-point-gapless
> ordered field. No multi-point interval (a,b) with a ~= b
> is without rationals. No (a,b) is a gap.
>
> The Least Upper Bound property eliminates single-point gaps.
>
> One important consequence of the elimination of single-point
> gaps is that we can formally describe and reason about
> Achilles chasing the tortoise along a _continuous_ path.
> Well, we could use the same definitions _without_ LUB, but they
> won't mean what we want them to mean. The path Achilles takes
> can be continuous _at every point_ but, if there are
> single-point gaps, he can execute arbitrarily large teleports
> across that gap. Sure, that wouldn't be continuous _if there_
> _was a point there_ but there isn't.
>
> That is the reason (one, at least) that we need the LUB
> property: so that "continuous at all the points"
> works out to "no teleporting".
> > (That the definition of continuous function applies to
> > domains that preserve a metric.)
> Continuous functions don't need to preserve a metric.
>
> Rotations and translations preserve the metric.
>
> Something along those lines that is true:
> The inverse f^-1 of a continuous function f maps open sets
> to open sets. We could say f^-1 preserves the topology.

Such a topology as about this points-in-a-line would
have two-point neighborhoods at endpoints and
three-point neighborhoods throughout the line.

Not sure "continuous functions don't preserve a metric".
I'll agree they're perhaps not affine, where they're also
transformations and make a mapping.

I.e. they define the reversible/invertible mapping the metric.

I thank you though, about the affine.

The continuous functions don't so much care that the image
is the rationals or complete ordered field or as about pretty
much any other topology with a metric.

On Sunday, August 29, 2021 at 12:10:22 PM UTC-7, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 11:24:51 AM UTC-7, Jim Burns wrote:
> > On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> > > On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> > > Jim Burns wrote:
> >
> > >> To borrow a term from art, a gap is _negative space_
> > >> It is an interval in which points are NOT.
> > >> https://en.wikipedia.org/wiki/Negative_space
> > >>
> > >> In order for there to be a point next to zero,
> > >> there must be negative space next to zero, and then the point.
> > >> If there is no negative space, there is no next point.
> > >> If there is no negative space, there is no gap.
> > >
> > > Least Upper Bound is the usual axiom attached to the
> > > ordered field to result the gapless complete ordered field.
> > Here, I'm using "gap" in a more restricted sense than I have
> > elsewhere. Call them "multi-point gaps".
> >
> > The rationals are already sufficient for a multi-point-gapless
> > ordered field. No multi-point interval (a,b) with a ~= b
> > is without rationals. No (a,b) is a gap.
> >
> > The Least Upper Bound property eliminates single-point gaps.
> >
> > One important consequence of the elimination of single-point
> > gaps is that we can formally describe and reason about
> > Achilles chasing the tortoise along a _continuous_ path.
> > Well, we could use the same definitions _without_ LUB, but they
> > won't mean what we want them to mean. The path Achilles takes
> > can be continuous _at every point_ but, if there are
> > single-point gaps, he can execute arbitrarily large teleports
> > across that gap. Sure, that wouldn't be continuous _if there_
> > _was a point there_ but there isn't.
> >
> > That is the reason (one, at least) that we need the LUB
> > property: so that "continuous at all the points"
> > works out to "no teleporting".
> > > (That the definition of continuous function applies to
> > > domains that preserve a metric.)
> > Continuous functions don't need to preserve a metric.
> >
> > Rotations and translations preserve the metric.
> >
> > Something along those lines that is true:
> > The inverse f^-1 of a continuous function f maps open sets
> > to open sets. We could say f^-1 preserves the topology.
> Such a topology as about this points-in-a-line would
> have two-point neighborhoods at endpoints and
> three-point neighborhoods throughout the line.
>
> Not sure "continuous functions don't preserve a metric".
> I'll agree they're perhaps not affine, where they're also
> transformations and make a mapping.
>
> I.e. they define the reversible/invertible mapping the metric.
>
> I thank you though, about the affine.
>
> The continuous functions don't so much care that the image
> is the rationals or complete ordered field or as about pretty
> much any other topology with a metric.

https://en.wikipedia.org/wiki/Natural_topology

On 8/29/2021 1:59 PM, FredJeffries wrote:
> On Friday, August 27, 2021 at 4:14:36 AM UTC-7, WM wrote:
>> Jim Burns schrieb am Donnerstag, 26. August 2021 um 23:17:48 UTC+2:
>
>>> If you can prove arithmetic is inconsistent, go ahead.
>> Arithmetic is consistent.
>
> Proof by Casting a Spell
>

agree, spoofs are proofs without the r and an s.

Arithmetic is counting rocks, and adding, and subtracting, and multiplying, and division. noting else.

Arithmetic is not geometry, not algebra, not set theory, not calculus, not vector fields, not...

William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>
> > What is next to zero? Note that "nothing" is synonymous to "gap".
> Nope
> A gap next to 0 is an interval [0,x] (note x must be a real number

The end of the gap is a real number x but this end is not fixed. It can be decreased. Neverthelss the dark gap remains.

> that does not contain an element of U. There is no such interval so there is no gap. So there is neither a gap nor a point next to 0.

That is a logical impossibility. There is either a definable point or no definable point next to zero. Undefinable points may be called gaps.

Regards, WM

On Sunday, August 29, 2021 at 9:43:07 PM UTC+2, WM wrote:
> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> >
> > There is no such interval so there is no gap. So there is neither a gap nor a point next to 0.
> >
> That is a logical impossibility.

Actually, the contrary is the case.

It's a logical necessity (if we presuppose the axioms of classical mathematics).

See: https://en.wikipedia.org/wiki/Classical_mathematics

On Sunday, August 29, 2021 at 12:39:51 PM UTC-7, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 12:10:22 PM UTC-7, Ross A. Finlayson wrote:
> > On Sunday, August 29, 2021 at 11:24:51 AM UTC-7, Jim Burns wrote:
> > > On 8/29/2021 1:33 PM, Ross A. Finlayson wrote:
> > > > On Sunday, August 29, 2021 at 10:06:47 AM UTC-7,
> > > > Jim Burns wrote:
> > >
> > > >> To borrow a term from art, a gap is _negative space_
> > > >> It is an interval in which points are NOT.
> > > >> https://en.wikipedia.org/wiki/Negative_space
> > > >>
> > > >> In order for there to be a point next to zero,
> > > >> there must be negative space next to zero, and then the point.
> > > >> If there is no negative space, there is no next point.
> > > >> If there is no negative space, there is no gap.
> > > >
> > > > Least Upper Bound is the usual axiom attached to the
> > > > ordered field to result the gapless complete ordered field.
> > > Here, I'm using "gap" in a more restricted sense than I have
> > > elsewhere. Call them "multi-point gaps".
> > >
> > > The rationals are already sufficient for a multi-point-gapless
> > > ordered field. No multi-point interval (a,b) with a ~= b
> > > is without rationals. No (a,b) is a gap.
> > >
> > > The Least Upper Bound property eliminates single-point gaps.
> > >
> > > One important consequence of the elimination of single-point
> > > gaps is that we can formally describe and reason about
> > > Achilles chasing the tortoise along a _continuous_ path.
> > > Well, we could use the same definitions _without_ LUB, but they
> > > won't mean what we want them to mean. The path Achilles takes
> > > can be continuous _at every point_ but, if there are
> > > single-point gaps, he can execute arbitrarily large teleports
> > > across that gap. Sure, that wouldn't be continuous _if there_
> > > _was a point there_ but there isn't.
> > >
> > > That is the reason (one, at least) that we need the LUB
> > > property: so that "continuous at all the points"
> > > works out to "no teleporting".
> > > > (That the definition of continuous function applies to
> > > > domains that preserve a metric.)
> > > Continuous functions don't need to preserve a metric.
> > >
> > > Rotations and translations preserve the metric.
> > >
> > > Something along those lines that is true:
> > > The inverse f^-1 of a continuous function f maps open sets
> > > to open sets. We could say f^-1 preserves the topology.
> > Such a topology as about this points-in-a-line would
> > have two-point neighborhoods at endpoints and
> > three-point neighborhoods throughout the line.
> >
> > Not sure "continuous functions don't preserve a metric".
> > I'll agree they're perhaps not affine, where they're also
> > transformations and make a mapping.
> >
> > I.e. they define the reversible/invertible mapping the metric.
> >
> > I thank you though, about the affine.
> >
> > The continuous functions don't so much care that the image
> > is the rationals or complete ordered field or as about pretty
> > much any other topology with a metric.
> https://en.wikipedia.org/wiki/Natural_topology

My point here is that though the usual "must be no smallest"
and "must be next largest" are at odds, that, when there are
both structures then each has all the usual features of what
makes for modeling each other's natural topologies.

That a vector space makes a basis for a topology,
and that ran(EF) is a vector space, with its range of iota-values.

"The natural/unit equivalency function that best relates
integers zero to infinity and reals zero through one is a
drawing of a line."

Once a model of real numbers that is iota-values is equipped
with all the other usual features what are handles for analysis,
there is implemented a neat extensionality what all the results
go both ways for either.

I.e. the length assignment for ran(f) the iota-values, pretty much
stands up as a default metric along with all the other usual naive
properties why this object is more primitive, and primary, in theory,
than needs require after all of topology the ultrafilter, or a countable
basis in a space for a characterization of real regions, that it's the
default natural model as for example by time of constant motion or change.

Or, besides being fundamentally at odds,
models of continuity are reflective and replete.

Then I have written this up and about its topologies
the contiguous points "in" a line vis-a-vis the
ordered field's points "on" a line, why each are
models of continuity, or here continuous domains.

Crank Wolfgang Mueckenheim, aka WM wrote:
> Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
>> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
>
>> There is only one Grossone it's Yaroslav Sergeyev's.
>>
> Yes.

Reminder:

---
Following the publication of an article by Sergeyev in EMS Surveys in
Mathematical Sciences, the editors published the following
clarification:

Statement of the editorial board

We deeply regret that this article appears in this issue of the EMS
Surveys in Mathematical Sciences.

It was a serious mistake to accept it for publication. Owing to an
unfortunate error, the entire processing of the paper, including the
decision to accept it, took place without the editorial board being
aware of what was happening. The editorial board unanimously
dissociates itself from this decision. It is not representative of the
very high level that we expect to see in our journal, which can be
assessed from all other papers that we have published.

Both editors-in-chief have assumed responsibility for these
mistakes and resigned from their position. Having said that, we add
that this journal would not exist without their dedication and years of
hard work, and we wish to register our thanks to them.
---

On Sunday, August 29, 2021 at 10:06:42 PM UTC+2, Python wrote:

> "Both editors-in-chief have assumed responsibility for these
> mistakes and resigned from their position."

If only Mückenheim would resign from his position.

On 8/29/2021 11:35 AM, WM wrote:
> Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
>> On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:
>
>>> There is nothing between the point next to zero and <bla>
>>
>> There is no "point next to zero",
>
> Then there is a gap. Then the real line is discontinuous. That is possible.
>
> Regards, WM
>

Wrong, no gap, no point. The real line is continuous.

You do not understand the basics, because you never took a class in the basics.

On 8/29/2021 11:38 AM, WM wrote:
> Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
>> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
>
>> There is only one Grossone it's Yaroslav Sergeyev's.
>>
> Yes.
>
>> The, "point next to zero", has about as much definition, as "point at infinity".
>
> No, point next to omega would be the analogy.

No, there are no points "next to" other point

EZ proof

call A, point A
call B, Point B

assume A is "next to" B
however (A+B)/2 is closer to A than B is
Assumption if False.

WM's math is Potato Math.

>
> Regards, WM
>

On 8/29/2021 3:20 PM, Greg Cunt wrote:
> On Sunday, August 29, 2021 at 10:06:42 PM UTC+2, Python wrote:
>
>> "Both editors-in-chief have assumed responsibility for these
>> mistakes and resigned from their position."
>
> If only Mückenheim would resign from his position.
>

all his students think and say the same thing.

On Sunday, August 29, 2021 at 4:43:07 PM UTC-3, WM wrote:
> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> > On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
> >
> > > What is next to zero? Note that "nothing" is synonymous to "gap".
> > Nope
> > A gap next to 0 is an interval [0,x] (note x must be a real number
> The end of the gap is a real number x

and a "gap", like a set, is not something that changes.

--
William Hughes

Serg io wrote:
> On 8/29/2021 3:20 PM, Greg Cunt wrote:
>> On Sunday, August 29, 2021 at 10:06:42 PM UTC+2, Python wrote:
>>
>>> "Both editors-in-chief have assumed responsibility for these
>>> mistakes and resigned from their position."
>>
>> If only Mückenheim would resign from his position.
>>
>
> all his students think and say the same thing.

Unfortunately (shame), Hochschule Augsburg considers teaching fallacies
in a math course to be a question of "academic freedom".

On 8/27/2021 7:27 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 27. August 2021 um 00:20:04 UTC+2:

>> Each unit fraction 1/k is not _next to_ 0
>> because 0 < 1/(k+1) < 1/k
>
> The cursor moving from 0 to 1 will encounter
> some unit fraction first.

The cursor can only encounter a non-unit-fraction unit fraction
first. Any unit-fraction unit fraction will not be first.

>> The rationals, in their standard order, cannot be stepped
>> one-by-one from any rational to any other rational.
>> This is equivalent to
>> no rational is _next to_ any other rational.
>
> The reason is that between two definble rationals,
> there are dark points.

No, dark points are not present in the reason.

| Suppose that q can be stepped to from p, one-by-one,
| skipping no rational r, p < r < q.
| Then these rationals from p to q are _steppable_
| Being steppable, for any split B, A, there is a
| crossing-pair p',q', p' in B, q' in A and no other
| rational between them.
| And there's the rub.
| For any rationals p',q', there is r' = (p' + q')/2
| and p' < r' < q'.
| r' is (hypothetically) not in B and not in A,
| but p < r' < q.
| Contradiction.
| | Therefore, q cannot be stepped to from p, one-by-one,
| skipping no rational r, p < r < q.

Dark points are not present in the reason.
What it means to be "able to step one-by-one" is present.
That (p' + q')/2 is between p' and q' is present.

>> This is the opposite of how it is with naturals.
>
> No, most natural are dark too.

The opposites being compared there have nothing to do
with dark numbers.

Two naturals in their standard order are _always_ able to be
stepped, one-by-one, from one to the other.

Two rationals in their standard order are _never_ able to be
stepped, one-by-one, from one to the other.

On 8/27/2021 10:23 AM, WM wrote:
> Serg io schrieb
> am Freitag, 27. August 2021 um 16:18:14 UTC+2:

>> Cantor enumerates all rationals.
>
> Why doesn't he start with all fractions n/1?

There are infinitely-many fractions n/1.
Starting with all n/1 first would make all the other
fractions inaccessible to one-by-one stepping.

Since the claim that that _at least one_ way to order
the rationals has them all (like the naturals) accessible
to one-by-one stepping (in that non-standard order)
from any other rationals,
starting with all n/1 is not a counter-example.
It is merely not an example.

But there are plenty of examples.
We didn't need yours, anyway.