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<span class="kw">let </span><font color="Maroon" title="c1">m1</font>, <font color="Maroon" title="c2">m2</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c2</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"> ( <font color="Maroon" title="c1">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">c2</font> &amp; <font color="Maroon" title="c1">m1</font>,<font color="Maroon" title="c2">m2</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies  ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( (  for <font color="Olive" title="b4">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b4">x</font>,<font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c4">c1</font>
 <span class="kw">and </span><br/><a NAME="E4:61"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">c2</font>
 <span class="kw">and </span><br/><a NAME="E5:61"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c1">m1</font>,<font color="Maroon" title="c2">m2</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( (  for <font color="Olive" title="b4">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b4">x</font>,<font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">d2</font> = <font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font>;<br/>
<a NAME="E6:61"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E2:61"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">d1</font> = <font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font>;<br/>
<a NAME="E7:61"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E1:61"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/>
<a NAME="E8:61"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>,<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="int_4.html#E52"><span class="lab"><font color="Green" title="E37">Lm12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E10:61"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <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="ref" href="int_4.html#T15" target="_self" title="INT_4:th.15">Th15</a></span>;<br/>
<a NAME="E11:61"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a></span>;<br/>
<a NAME="E12:61"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E13:61"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E14:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<a NAME="E15:61"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E2:61"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:61"><span class="lab"><font color="Green" title="E53">A12</font></span></a></span>;<br/>
<a NAME="E16:61"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a></span>;<br/>
<a NAME="E17:61"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E18:61"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>,<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="int_4.html#E52"><span class="lab"><font color="Green" title="E37">Lm12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E20:61"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <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="ref" href="int_4.html#T15" target="_self" title="INT_4:th.15">Th15</a></span>;<br/>
<a NAME="E21:61"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E22:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<a NAME="E23:61"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>,<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">e</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:61_1"/>
 not <font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>,<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E4:61_1"/>
<font color="Maroon" title="c11">e</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E5:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A23</font></span>: 
<font color="Maroon" title="c11">e</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></span>;<br/>
<a NAME="E6:61_1"/>
<font color="Maroon" title="c11">e</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E7:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></span>;<br/>
<a NAME="E8:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E64">A23</font></span></a>, <a class="ref" href="int_2.html#T22" title="INT_2:th.22">INT_2:22</a></span>;<br/>
<a NAME="E9:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:61_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E63">A22</font></span></a>, <a class="ref" href="wsierp_1.html#T15" title="WSIERP_1:th.15">WSIERP_1:15</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">w</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E25:61"/><span class="lab"><font color="Green" title="E63">A24</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <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="ref" href="int_4.html#T15" target="_self" title="INT_4:th.15">Th15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">w</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( (  for <font color="Olive" title="b3">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( (  for <font color="Olive" title="b2">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b2">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">s</font>
; <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">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E26:61"/><span class="lab"><font color="Green" title="E64">A25</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E1:61"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E60">A19</font></span></a></span>;<br/>
<a NAME="E27:61"/>
 for <font color="Olive" title="b1">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</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">and </span><br/><a NAME="E2:61_2"/><span class="lab"><font color="Green" title="E66">A27</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:61_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E21:61"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="txt" href="#E16:61"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="txt" href="#E17:61"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="txt" href="int_4.html#E50"><span class="lab"><font color="Green" title="E35">Lm10</font></span></a></span>;<br/>
<a NAME="E4:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="int_4.html#T7" target="_self" title="INT_4:th.7">Th7</a></span>;<br/>
<a NAME="E5:61_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <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_4.html#E50"><span class="lab"><font color="Green" title="E35">Lm10</font></span></a></span>;<br/>
<a NAME="E6:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E10:61"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E26:61"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="int_4.html#T26" target="_self" title="INT_4:th.26">Th26</a></span>;<br/>
<a NAME="E7:61_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" title="EQREL_1:th.18">EQREL_1:18</a></span>;<br/>
<a NAME="E8:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">r</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<a NAME="E9:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A28</font></span>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E10:61_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E13:61"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E12:61"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E2:61_2"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="int_4.html#E50"><span class="lab"><font color="Green" title="E35">Lm10</font></span></a></span>;<br/>
<a NAME="E11:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="int_4.html#T7" target="_self" title="INT_4:th.7">Th7</a></span>;<br/>
<a NAME="E12:61_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <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_4.html#E50"><span class="lab"><font color="Green" title="E35">Lm10</font></span></a></span>;<br/>
<a NAME="E13:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c12">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="int_4.html#T26" target="_self" title="INT_4:th.26">Th26</a></span>;<br/>
<a NAME="E14:61_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c10">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T18" title="EQREL_1:th.18">EQREL_1:18</a></span>;<br/>
<a NAME="E15:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c10">s</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<a NAME="E16:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A29</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E17:61_2"/>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E25:61"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="txt" href="int_4.html#E50"><span class="lab"><font color="Green" title="E35">Lm10</font></span></a></span>;<br/>
<a NAME="E18:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_2"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="int_4.html#E9"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<a NAME="E19:61_2"/>
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T2" title="INT_2:th.2">INT_2:2</a></span>;<br/>
<a NAME="E20:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_2"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="txt" href="int_4.html#E9"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/>
<a NAME="E21:61_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c12">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <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">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:61"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="txt" href="#E18:61_2"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="int_4.html#E53"><span class="lab"><font color="Green" title="E38">Lm13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:61_2"/>
<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E28:61"/>
( (  for <font color="Olive" title="b1">x</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <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="#E10:61"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E25:61"><span class="lab"><font color="Green" title="E63">A24</font></span></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/>


</div>
