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<b>reconsider </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="scmring2.html#D1" title="SCMRING2:def.1">SCMRING2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font>;<br/>
<b>set </b><font color="Maroon" title="c6">f</font> = the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<i><font color="Green" title="E32">A1</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2" title="SCMRING1:func.2">SCM-OK</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="scmring2.html#D1" title="SCMRING2:def.1">SCMRING2:def 1</a></i>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E3:46_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E4:46_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E5:46_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 ;<br/>

<a NAME="E2:46_1_1_2"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="scmring4.html#E3:46_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_1_1_2"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="scmring4.html#E2:46_1_1_2"><i><font color="Green" title="E35">A4</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="scmring3.html#T1" title="SCMRING3:th.1">SCMRING3:1</a></i>
;<br/>
<b>hence </b><a NAME="E5:46_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="scmring4.html#E3:46_1_1_2"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_1_1"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">sproduct</a> the <a href="ami_1.html#U5" title="AMI_1:sel.5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmring4.html#E2:46_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="scmring4.html#E3:46_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="scmring4.html#E4:46_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="ami_1.html#D16" title="AMI_1:def.16">AMI_1:def 16</a></i>;<br/>
<b>thus </b><a NAME="E7:46_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c3">r</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">FinPartState</a> of  <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="scmring4.html#E6:46_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="ami_1.html#D24" title="AMI_1:def.24">AMI_1:def 24</a></i>;<br/>


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