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<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">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">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="int_1.html#R1">are_congruent_mod</a> 2
 ;<br/><a NAME="E2:77_1"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font>,1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="int_1.html#R1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T42">INT_1:42</a></i>;<br/><a NAME="E3:77_1"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span>,2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="int_1.html#R1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T35">INT_1:35</a></i>;<br/><a NAME="E4:77_1"/><b>then </b><i><font color="Green">E24</font></i>: 
1,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="int_1.html#R1">are_congruent_mod</a> 2
 <b>by </b><i><a class="ref" href="int_1.html#T40">INT_1:40</a></i>;<br/><a NAME="E5:77_1"/>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span>,0 <a href="int_1.html#R1">are_congruent_mod</a> 2
 <b>by </b><i/>;<br/><a NAME="E6:77_1"/><b>then </b>
1,0 <a href="int_1.html#R1">are_congruent_mod</a> 2
 <b>by </b><i>, <a class="ref" href="int_1.html#T36">INT_1:36</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E8:77_1"/><i><font color="Green">E25</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D3">INT_1:def 3</a></i>;<br/><b>thus </b><a NAME="E9:77_1"/>
contradiction
 <b>by </b><i><a class="ref" href="ami_3.html#T9" target="_self">Th9</a>, <a class="ref" href="int_1.html#T22">INT_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:77"/>
 <a href="ami_3.html#K15" target="_self">dl.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K16" target="_self">il.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="int_1.html#T32">INT_1:32</a></i>;<br/>


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