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<b>thus </b><a NAME="E1:20_1_1"/>
( <font color="Maroon">IT</font> is <a href="funct_1.html#V2">one-to-one</a> implies for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">IT</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

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



<b>assume </b><a NAME="E2:20_1_1_1"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> )
 ;<br/>

<a NAME="E3:20_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1_1_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:20_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E3:20_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x1</font>, <font color="Maroon">y'</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:20_1_1"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2"/>
( <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> or <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_1_1"/>
( <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y'</font> or <font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x'</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_2"/>
<font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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