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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</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">l</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> <br/> for <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Olive" title="b3">k2</font></span>)</span>,<font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">l</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">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Olive" title="b2">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">k2</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">  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">t</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">l</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D14" title="AMI_1:def.14">AMI_1:def 14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c2">l</font> as    <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>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">il1</font> = <font color="Maroon" title="c2">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="ami_1.html#D11" title="AMI_1:def.11">AMI_1:def 11</a></span>;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E16">X</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">il1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="ami_1.html#K22" title="AMI_1:func.22">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span></span>)</span> is  the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  <a href="funct_1.html#V5" title="FUNCT_1:attr.5">-compatible</a>  <a href="ami_1.html#NM3" title="AMI_1:NM.3">PartState</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
 
;<br/>
<a NAME="E6:24"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">il1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">u</font> = <font color="Maroon" title="c5">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<font color="Maroon" title="c2">l</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c10">il1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">X</font></span></a>, <a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c13">s</font></span>)</span>,<font color="Maroon" title="c13">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">and </span><br/><a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font>
 ;<br/><a NAME="E6:24_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 ex <font color="Olive" title="b1">m1</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>  st <br/>( <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font> &amp;  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c4">k2</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">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D20" title="SCMPDS_2:def.20">SCMPDS_2:def 20</a></span>;<br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E21">Y</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T150" title="AMI_1:th.150">AMI_1:150</a></span>;<br/><a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E22">y</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c13">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T15" title="AMI_1:th.15">AMI_1:15</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E21">Y</font></span></a>, <a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E22">y</font></span></a>, <a class="ref" href="ami_1.html#T131" title="AMI_1:th.131">AMI_1:131</a></span>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E1:24_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="scmpds_2.html#T67" title="SCMPDS_2:th.67">SCMPDS_2:67</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_2"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c13">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E21">Y</font></span></a>, <a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E22">y</font></span></a>, <a class="ref" href="ami_1.html#T131" title="AMI_1:th.131">AMI_1:131</a></span>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="scmpds_2.html#T67" title="SCMPDS_2:th.67">SCMPDS_2:67</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:24_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:24"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c12">x</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="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">l</font>
 ; <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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">u1</font> = <font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">u2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <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="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/><a NAME="E3:24_2_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></span>;<br/><a NAME="E4:24_2_1"/>
<font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>
<br/>.= 
<font color="Maroon" title="c11">u</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><span class="lab"><font color="Green" title="E21">A12</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font> = 
<font color="Maroon" title="c14">u2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a>, <a class="ref" href="scmpds_2.html#T52" title="SCMPDS_2:th.52">SCMPDS_2:52</a></span>
<br/>.= 
<font color="Maroon" title="c11">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a>, <a class="ref" href="scmpds_2.html#T52" title="SCMPDS_2:th.52">SCMPDS_2:52</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E7:24_2_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></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="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</a></span>;<br/><a NAME="E8:24_2_1"/><span class="lab"><font color="Green" title="E23">X</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T150" title="AMI_1:th.150">AMI_1:150</a></span>;<br/><a NAME="E9:24_2_1"/><span class="lab"><font color="Green" title="E24">Y</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T15" title="AMI_1:th.15">AMI_1:15</a></span>;<br/><a NAME="E10:24_2_1"/>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_1_3_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font> or <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_3_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 ; <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="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_2_1_3_1_1"/>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></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="#E7:24_2_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_3_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 ; <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="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:24_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_2_1_3_1_2"/>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></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="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c14">u2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="txt" href="#E1:24_2_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E6:24_2_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="scmpds_2.html#T67" title="SCMPDS_2:th.67">SCMPDS_2:67</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span>,<font color="Maroon" title="c14">u2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_2_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E6:24_2_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E9:24_2_1"><span class="lab"><font color="Green" title="E24">Y</font></span></a>, <a class="ref" href="ami_1.html#T131" title="AMI_1:th.131">AMI_1:131</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:24_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_2_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E6:24_2_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E8:24_2_1"><span class="lab"><font color="Green" title="E23">X</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_2"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c12">x</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="Maroon" title="c4">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">l</font></span>)</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" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c13">u1</font> = <font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">u2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <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="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/><a NAME="E3:24_2_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></span>;<br/><a NAME="E4:24_2_2"/>
<font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A16</font></span>: <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>
<br/>.= 
<font color="Maroon" title="c11">u</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>
<br/>.= 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E6:24_2_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</a></span>;<br/><a NAME="E7:24_2_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_2_2_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font> or <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_2_2_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 ; <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="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_2_2_2_1_1"/>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E6:24_2_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_2_2_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font>
 ; <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="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:24_2_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_2_2_2_1_2"/>
<font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">u2</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon" title="c3">k1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E23">A19</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font> = 
<font color="Maroon" title="c14">u2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="scmpds_3.html#K1" title="SCMPDS_3:func.1">.--&gt;</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a>, <a class="ref" href="scmpds_2.html#T52" title="SCMPDS_2:th.52">SCMPDS_2:52</a></span>
<br/>.= 
<font color="Maroon" title="c11">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="funct_4.html#T88" title="FUNCT_4:th.88">FUNCT_4:88</a>, <a class="ref" href="scmpds_2.html#T52" title="SCMPDS_2:th.52">SCMPDS_2:52</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></span>
<br/>.= 
<font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E9:24_2_2"/><span class="lab"><font color="Green" title="E24">X</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T150" title="AMI_1:th.150">AMI_1:150</a></span>;<br/><a NAME="E10:24_2_2"/><span class="lab"><font color="Green" title="E25">Y</font></span>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u2</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T15" title="AMI_1:th.15">AMI_1:15</a></span>;<br/><a NAME="E11:24_2_2"/>
 ex <font color="Olive" title="b1">m1</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>  st <br/>( <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c14">u2</font> &amp;  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c14">u2</font>,<font color="Maroon" title="c4">k2</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">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D20" title="SCMPDS_2:def.20">SCMPDS_2:def 20</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c14">u2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</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="txt" href="#E1:24_2_2"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E8:24_2_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E7:24_2_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="scmpds_2.html#T67" title="SCMPDS_2:th.67">SCMPDS_2:67</a></span>
<br/>.= 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c14">u2</font></span>)</span>,<font color="Maroon" title="c14">u2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_2_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E8:24_2_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E10:24_2_2"><span class="lab"><font color="Green" title="E25">Y</font></span></a>, <a class="ref" href="ami_1.html#T131" title="AMI_1:th.131">AMI_1:131</a></span>
;<br/><span class="kw">hence </span><a NAME="E13:24_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">k1</font> <a href="scmpds_2.html#K8" title="SCMPDS_2:func.8">&lt;&gt;0_goto</a> <font color="Maroon" title="c4">k2</font></span>)</span>,<font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_2_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E8:24_2_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E9:24_2_2"><span class="lab"><font color="Green" title="E24">X</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
