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<span class="kw">let </span><font color="Maroon" title="c13">s</font> be   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="ami_1.html#V12" title="AMI_1:attr.12">-started</a>  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Olive" title="b1">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b3">y</font> &amp;  ex <font color="Olive" title="b4">k</font> being   <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>  st <font color="Olive" title="b1">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Olive" title="b1">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b4">k</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">P</font> be  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">P</font> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">y</font> &amp;  ex <font color="Olive" title="b3">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b3">k</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="ami_4.html#K2" title="AMI_4:func.2">Euclide-Algorithm</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">y</font> &amp;  ex <font color="Olive" title="b3">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b3">k</font></span>)</span> )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> $1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span>)</span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <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">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> $1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font> &amp;  ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b1">k</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">and </span><br/><a NAME="E3:15"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">and </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">y</font>
 <span class="kw">and </span><br/><a NAME="E5:15"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font> &amp;  ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b1">k</font></span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E18">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) ) )</span><br/><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</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="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:15_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>) )</span><br/><span class="kw">hence </span><a NAME="E3:15_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="ami_4.html#E15"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>)</span><br/><a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E5:15_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1) = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p3">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E2:15_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="ami_4.html#E15"><span class="lab"><font color="Green" title="E11">Lm5</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c17">k</font>) <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="int_2.html#T49" title="INT_2:th.49">INT_2:49</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x9</font> = <font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c18">y9</font> = <font color="Maroon" title="c16">y</font> 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="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<a NAME="E8:15"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c18">y9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A12</font></span>: <font color="Maroon">H<sub>2</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c17">x9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<span class="lab"><font color="Green" title="E21">A13</font></span>: <font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c18">y9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<a NAME="E11:15"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">k</font> being    <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"> such that </span><br/><a NAME="E13:15"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c19">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x9</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">y9</font>
 <span class="kw">and </span><br/><a NAME="E14:15"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">k</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">from</span> <span class="lab"><a class="ref" href="newton.html#S1" title="NEWTON:sch.1">NEWTON:sch 1</a>(<a class="txt" href="#E11:15"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:15"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A7</font></span></a>);<br/></span>
<a NAME="E15:15"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/>
<a NAME="E16:15"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> 1</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="#E14:15"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/>
<a NAME="E17:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A18</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="ami_4.html#E14"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/>
<a NAME="E18:15"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="ami_4.html#E8"><span class="lab"><font color="Green" title="E4">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="ami_1.html#K11" title="AMI_1:func.11">Result</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="ami_1.html#T87" title="AMI_1:th.87">AMI_1:87</a></span>
<br/>.= 
<font color="Maroon" title="c15">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E15:15"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b1">k</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E20:15"/>
 ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c14">P</font> <a href="ami_1.html#R3" title="AMI_1:pred.3">halts_at</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <font color="Maroon" title="c14">P</font>,<font color="Maroon" title="c13">s</font>,<font color="Olive" title="b1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E18:15"><span class="lab"><font color="Green" title="E27">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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