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<span class="kw">let </span><font color="Maroon" title="c13">s</font> be   <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">k</font>, <font color="Olive" title="b3">i</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  <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="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 holds <br/> <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>,<span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="b2">k</font></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">k</font>, <font color="Olive" title="b2">i</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  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 holds <br/> <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="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><br/>



<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E10">A1</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">k</font>, <font color="Olive" title="b2">i</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  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 holds <br/> <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="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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><br/>

<span class="kw">let </span><font color="Maroon" title="c15">k</font>, <font color="Maroon" title="c16">i</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"> (  <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="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies  <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="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Maroon" title="c15">k</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:12"/>
 <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="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Maroon" title="c15">k</font></span><br/>

<a NAME="E3:12"/><span class="kw">then </span>
<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="Maroon" title="c15">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E10">A1</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><a NAME="E4:12"/>
 <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="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T89" title="AMI_1:th.89">AMI_1:89</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</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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