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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">i'</font>, <font color="Maroon" title="c7">j'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c6">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font>;<br/><b>thus </b><a NAME="E2:21_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<a NAME="E2:21_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i'</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i'</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E1:21_1_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel17.html#E1:21_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="waybel17.html#E1:21_1_1_1"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:21_1_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c6">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">k</font>
 ;<br/>
<a NAME="E5:21_1_1_1"/><i><font color="Green" title="E17">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c7">j'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_2"/>
<font color="Maroon" title="c7">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">k1</font>
 ;<br/>

<a NAME="E2:21_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">k1</font>
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E4:21_1_1_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E5:21_1_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E7:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E3:21_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E4:21_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i'</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i'</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E1:21_1_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E3:21_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel17.html#E1:21_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="waybel17.html#E3:21_1_1"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:21_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">j'</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_3"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">j'</font>]
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">kl</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:21_1_1_3"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c7">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kl</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="scheme1.html#T1" title="SCHEME1:th.1">SCHEME1:1</a></i>;<br/>
<b>thus </b><a NAME="E4:21_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E4:21_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="waybel17.html#E3:21_1_1_3"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="waybel17.html#K1" title="WAYBEL17:func.1">,...</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="waybel17.html#E1:21"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel17.html#E5:21_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="waybel17.html#D1" target="_self" title="WAYBEL17:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E7:21_1_1"/>
<span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_2"/>
( ( <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) &amp; ( <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) )
 <b>by </b><i><a class="ref" href="waybel17.html#T5" target="_self" title="WAYBEL17:th.5">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
