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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span><br/>

<a NAME="E2:21"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#T6" target="_self" onmouseover="rs('ntalgo_1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">s1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">s3</font> = <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E6:21"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T50" onmouseover="rs('int_1/T50')" onmouseout="rh()">INT_1:50</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A4</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="int_1.html#T73" onmouseover="rs('int_1/T73')" onmouseout="rh()">INT_1:73</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">I0</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A5</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#T1" target="_self" onmouseover="rs('ntalgo_1/T1')" onmouseout="rh()">Th1</a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_3"/>
( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_1"><span class="kw">suppose </span></a><a NAME="E1:21_3_1"/>
<span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E3:21_3_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp;  <a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#D3" target="_self" onmouseover="rs('ntalgo_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T61" onmouseover="rs('nat_d/T61')" onmouseout="rh()">NAT_D:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">z</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#T1" target="_self" onmouseover="rs('ntalgo_1/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:21_3_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_2"><span class="kw">suppose </span></a><a NAME="E1:21_3_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ntalgo_1.html#K3" title="NTALGO_1:func.3">ALGO_INVERSE</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#D3" target="_self" onmouseover="rs('ntalgo_1/D3')" onmouseout="rh()">Def3</a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s3</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s3</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ntalgo_1.html#T1" target="_self" onmouseover="rs('ntalgo_1/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
1 <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="nat_d.html#T67" onmouseover="rs('nat_d/T67')" onmouseout="rh()">NAT_D:67</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

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