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<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>;<br/>
<i><font color="Green">E45</font></i>:  <a href="relat_1.html#K1">dom</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmring1.html#K2">SCM-OK</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E3:46_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E4:46_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E5:46_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<a NAME="E2:46_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:46_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="ami_1.html#K3">ObjectKind</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="scmring3.html#T1">SCMRING3:1</a></i>
;<br/>
<b>hence </b><a NAME="E5:46_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:46_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7">sproduct</a> the <a href="ami_1.html#U5">Object-Kind</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="ami_1.html#D16">AMI_1:def 16</a></i>;<br/>
<b>thus </b><a NAME="E7:46_1_1"/>
<font color="Maroon">a</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">r</font> is    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="ami_1.html#D24">AMI_1:def 24</a></i>;<br/>


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