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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>, <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>;<br/><b>let </b><font color="Maroon">i'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><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> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font>;<br/><b>thus </b><a NAME="E2:21_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E47</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:21_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">E48</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i'</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E3:21_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">k</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="E4:21_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 ;<br/>
<a NAME="E5:21_1_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">j'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">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">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_1_2"/>
<font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">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#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, </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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E7:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E3:21_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E4:21_1_1"/><i><font color="Green">E54</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i'</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E3:21_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E5:21_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">j'</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">kl</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="E3:21_1_1_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">kl</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="scheme1.html#T1">SCHEME1:1</a></i>;<br/>
<a NAME="E4:21_1_1_3"/>
<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">kl</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:21_1_1_3"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="waybel17.html#K1" target="_self">,...</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E7:21_1_1"/>
<span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) &amp; ( <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
