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<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>, <font color="Maroon">f2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_2.html#V3">bijective</a>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f2</font> is <a href="funct_2.html#V3">bijective</a>
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E5:28"/>
<font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f2</font>
 ;<br/>

<a NAME="E6:28"/><b>then </b><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>]
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:28"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">d</font>]
 <b>and </b><br/><a NAME="E9:28"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E10:28"/>
( <font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f2</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E11:28"/><b>then </b><i><font color="Green">E67</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">d2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:28"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">d2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:28"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E18:28"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E19:28"/><i><font color="Green">E71</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( ( <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) or ( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) or ( <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d2</font> ) or ( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E3:28_1_1"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:28_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_2"/>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</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"/>
not <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font>
 ;<br/>

<a NAME="E2:28_1_2_1"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3"><b>suppose </b></a><a NAME="E1:28_1_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d2</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_3"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_3_1"/>
not <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d1</font>
 ;<br/>

<a NAME="E2:28_1_3_1"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_3"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_4"><b>suppose </b></a><a NAME="E1:28_1_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d2</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_4"/><b>then </b>
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d2</font>
 <b>by </b><i/>;<br/><a NAME="E3:28_1_4"/><b>then </b>
<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:28_1_4"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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