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<span class="kw">let </span><font color="Maroon" title="c6">k1</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  not  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font> is  <a href="extpro_1.html#V2" title="EXTPRO_1:attr.2">halting</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c6">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font> is  <a href="extpro_1.html#V2" title="EXTPRO_1:attr.2">halting</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">n</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">loc</font> = <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 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>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><span class="kw"> such that </span><br/><a NAME="E4:77"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">loc</font>
 <span class="kw">and </span><br/><a NAME="E5:77"/>
 for <font color="Olive" title="b1">d</font> being    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds  <font color="Maroon" title="c9">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">d</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="ref" href="scmpds_2.html#T62" target="_self" title="SCMPDS_2:th.62">Th74</a></span>;<br/>
<a NAME="E6:77"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" title="ABSVALUE:th.4">ABSVALUE:4</a></span>;<br/>
<a NAME="E7:77"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k1</font>
 
;<br/>
<a NAME="E8:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <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="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/>
<a NAME="E9:77"/>
 ex <font color="Olive" title="b1">m</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 <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c9">s</font> &amp;  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> (<font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c6">k1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D18" target="_self" title="SCMPDS_2:def.18">Def20</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A4</font></span>: <span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font></span>)</span>,<font color="Maroon" title="c9">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="scmpds_2.html#T2" target="_self" title="SCMPDS_2:th.2">Th6</a>, <a class="ref" href="scmpds_2.html#T54" target="_self" title="SCMPDS_2:th.54">Th66</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<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="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <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="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">k1</font>

;<br/>
<span class="kw">assume </span><a NAME="E11:77"/>
 <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font> is  <a href="extpro_1.html#V2" title="EXTPRO_1:attr.2">halting</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E12:77"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <font color="Maroon" title="c6">k1</font></span>)</span>,<font color="Maroon" title="c9">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="scmpds_2.html#T2" target="_self" title="SCMPDS_2:th.2">Th6</a>, <a class="ref" href="extpro_1.html#D3" title="EXTPRO_1:def.3">EXTPRO_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:77"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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