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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c1">i</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">j</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Olive" title="b1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c2">j</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c3">m</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E2:14"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">j</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c3">m</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:14"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="int_5.html#D2" target="_self" title="INT_5:def.2">Def2</a></span>;<br/>
<a NAME="E5:14"/>
<font color="Maroon" title="c3">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E6:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_5.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" title="NAT_D:th.66">NAT_D:66</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="nat_d.html#T65" title="NAT_D:th.65">NAT_D:65</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:14"/>
<font color="Maroon" title="c2">j</font> <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#D2" target="_self" title="INT_5:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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