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<span class="kw">let </span><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</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"> ( (  for <font color="Olive" title="b1">y</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>   st <font color="Olive" title="b1">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="c2">p</font> holds <br/>( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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 implies ( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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>  implies  ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">z</font> ) ) &amp; (  <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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  implies <font color="Maroon" title="c3">s1</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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  ) ) ) &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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) ) &amp; (  for <font color="Olive" title="b1">y</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>   st <font color="Olive" title="b1">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="c2">p</font> holds <br/>( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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 implies ( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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>  implies  ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">z</font> ) ) &amp; (  <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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  implies <font color="Maroon" title="c4">s2</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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  ) ) ) &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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) ) implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1_2"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
 for <font color="Olive" title="b1">y</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>   st <font color="Olive" title="b1">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="c2">p</font> holds <br/>( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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 implies ( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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>  implies  ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">z</font> ) ) &amp; (  <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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  implies <font color="Maroon" title="c3">s1</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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  ) ) ) &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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> (  ex <font color="Olive" title="b1">y</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>  st <br/>( <font color="Olive" title="b1">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="c2">p</font> &amp;  not <!--HUMANRECFAILED-->( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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 implies ( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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>  implies  ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">z</font> ) ) &amp; (  <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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  implies <font color="Maroon" title="c4">s2</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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  ) ) ) &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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) ) or <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:20_1_2"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
 for <font color="Olive" title="b1">y</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>   st <font color="Olive" title="b1">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="c2">p</font> holds <br/>( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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 implies ( ( <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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>  implies  ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b2">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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">z</font> ) ) &amp; (  <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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  implies <font color="Maroon" title="c4">s2</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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  ) ) ) &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="c2">p</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">p</font> 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="kw">thus </span><a NAME="E4:20_1_2"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
  <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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1"><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="E1:20_1_2_1_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="c2">p</font>,<font color="Maroon" title="c5">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 or <span class="p1">(<span class="default"><a href="ntalgo_1.html#K2" title="NTALGO_1:func.2">ALGO_EXGCD</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">A8</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="c2">p</font>,<font color="Maroon" title="c5">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
 ; <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="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:20_1_2_1_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
  <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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_1_1_1"><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="E1:20_1_2_1_1_1_1_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="c2">p</font>,<font color="Maroon" title="c5">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="c2">p</font>,<font color="Maroon" title="c5">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="20_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A9</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="c2">p</font>,<font color="Maroon" title="c5">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"> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><a NAME="E2:20_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">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="c2">p</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:20_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>;<br/><a NAME="E3:20_1_2_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">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="c2">p</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a>  &amp; <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:20_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E1:20_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></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="20_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 <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="c2">p</font>,<font color="Maroon" title="c5">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"> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c3">s1</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="c2">p</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2_3</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E1:20_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></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>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_2_1_1_2"/><span class="lab"><font color="Green" title="E22">A12</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="c2">p</font>,<font color="Maroon" title="c5">y</font>)</span>)</span> <a href="mcart_1.html#K3" title="MCART_1:func.3">`3_3</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c3">s1</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:20_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></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>

</div><span class="kw">end;</span></div>


</div>
