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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">x</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">an</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c5">an</font>,<font color="Maroon" title="c1">a</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="gr_cy_3.html#E44"><span class="lab"><font color="Green" title="E23">Lm8</font></span></a></span>;<br/>
<a NAME="E5:45"/>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="int_3.html#T9" title="INT_3:th.9">INT_3:9</a></span>;<br/>
<a NAME="E6:45"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> is    <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">nb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:45"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c6">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">ai</font> = <font color="Maroon" title="c5">an</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E10:45"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c7">ai</font>,<font color="Maroon" title="c2">b</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">bi</font> = <font color="Maroon" title="c6">nb</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E12:45"/>
<font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T13" title="INT_3:th.13">INT_3:13</a></span>;<br/>
<a NAME="E13:45"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c8">bi</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E14:45"/><span class="kw">then </span>
<font color="Maroon" title="c5">an</font>,<font color="Maroon" title="c6">nb</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></span>;<br/>
<a NAME="E15:45"/><span class="kw">then </span>
<font color="Maroon" title="c5">an</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">nb</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="gr_cy_3.html#E45"><span class="lab"><font color="Green" title="E24">Lm9</font></span></a></span>;<br/>
<a NAME="E16:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">an</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nb</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E17:45"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T30" target="_self" title="GR_CY_3:th.30">Th30</a></span>;<br/>
<a NAME="E18:45"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T30" target="_self" title="GR_CY_3:th.30">Th30</a></span>;<br/>
<a NAME="E19:45"/>
<font color="Maroon" title="c5">an</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E20:45"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="pepin.html#T12" title="PEPIN:th.12">PEPIN:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:45"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E18:45"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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