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

<b>thus </b><a NAME="E1:2_1_2"/>
for <font color="Olive">m1</font>, <font color="Olive">m2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">m1</font> <a href="hidden.html#R1">=</a> 0 ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( <font color="Olive">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m1</font> holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m1</font> holds <br/><font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">m2</font> <a href="hidden.html#R1">=</a> 0 ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( <font color="Olive">m2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m2</font> holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m2</font> holds <br/><font color="Olive">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) ) holds <br/><font color="Olive">m1</font> <a href="hidden.html#R1">=</a> <font color="Olive">m2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_2_1"/><i><font color="Green">E70</font></i>: 
( (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> 0 ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> holds <br/><font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> 0 ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m2</font> holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r2</font> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m2</font> holds <br/><font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) ) ) )
 ;<br/>

<a NAME="E2:2_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font>
 
;<br/>
<a NAME="E3:2_1_2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:2_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E5:2_1_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:2_1_2_1"/>
<font color="Maroon">m2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:2_1_2_1"/>
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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


</div>
