<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">r1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">r2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:28_1_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">l</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">l</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> ) &amp; ( for <font color="Olive">l</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">l</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> ) implies <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>assume </b><a NAME="E2:28_1_2_1"/><i><font color="Green">E42</font></i>: 
( ( for <font color="Olive">l</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">l</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> ) &amp; ( for <font color="Olive">l</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">l</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E4:28_1_2_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="newton.html#K6" target="_self">!</a> </span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:28_1_2_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:28_1_2"/>
( not <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 ;<br/>


</div>
