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<span class="kw">let </span><font color="Maroon" title="c1">loc</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2: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"> for <font color="Olive" title="b1">I</font> being  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Olive" title="b1">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">loc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  holds <br/><font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">I</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">loc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  implies <font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">loc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">SS</font> =  <a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S2</font> =  <a href="valued_1.html#K61" title="VALUED_1:func.61">Shift</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E3:3"/>
not <font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:3"/><span class="kw">then </span>
<font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K61" title="VALUED_1:func.61">Shift</a> <span class="p2">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>;<br/>
<a NAME="E5:3"/><span class="kw">then </span>
<font color="Maroon" title="c1">loc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> where <font color="Olive" title="b1">l1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  : <font color="Olive" title="b1">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D12" title="VALUED_1:def.12">VALUED_1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">l1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:3"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c1">loc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:3"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c5">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
 ;<br/>
<a NAME="E9:3"/><span class="lab"><font color="Green" title="E8">y</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmnorm.html#T2" title="SCMNORM:th.2">SCMNORM:2</a></span>;<br/>
<a NAME="E10:3"/><span class="lab"><font color="Green" title="E9">x</font></span>: 
<span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/>
<a NAME="E11:3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmnorm.html#K2" title="SCMNORM:func.2">Stop</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a>, <a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a></span>;<br/>
<a NAME="E12:3"/><span class="kw">then </span>
<font color="Maroon" title="c5">l1</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">by</span> <span class="lab"><a class="txt" href="#E8:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E7:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E10:3"><span class="lab"><font color="Green" title="E9">x</font></span></a>, <a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E8">y</font></span></a>, <a class="ref" href="scmpds_4.html#T38" title="SCMPDS_4:th.38">SCMPDS_4:38</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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