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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 ;<br/><a NAME="E2:77_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font>,1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T42" title="INT_1:th.42">INT_1:42</a></i>;<br/><a NAME="E3:77_1"/><b>then </b>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font></span>)</span>,2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/><a NAME="E4:77_1"/><b>then </b><i><font color="Green" title="E51">A1</font></i>: 
1,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T40" title="INT_1:th.40">INT_1:40</a></i>;<br/><a NAME="E5:77_1"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i</font></span>)</span>,0 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="ami_3.html#T18" target="_self" title="AMI_3:th.18">Th18</a></i>;<br/><a NAME="E6:77_1"/><b>then </b>
1,0 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <b>by </b><i><a class="txt" href="ami_3.html#E4:77_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E8:77_1"/><i><font color="Green" title="E52">A2</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></i>;<br/><b>thus </b><a NAME="E9:77_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ami_3.html#E8:77_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="int_1.html#T22" title="INT_1:th.22">INT_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:77"/>
 <a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K16" title="AMI_3:func.16">il.</a> <font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="ref" href="int_1.html#T32" title="INT_1:th.32">INT_1:32</a></i>;<br/>


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