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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>

<a NAME="E1:11"/><i><font color="Green">E37</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) )
 ;<br/><a NAME="E2:11_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T18">SCMFSA_2:18</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E45</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E5:11_1"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:11_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:11_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> </span>}</span> ;<br/>
<a NAME="E3:11"/><i><font color="Green">E49</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>,<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="scmfsa_7.html#S1" target="_self">SCMFSA_7:sch 1</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E5:11"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:11"/><i><font color="Green">E51</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E7:11"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E9:11"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>thus </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K1" target="_self">Load</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">T</font>
<b>by </b><i>, , <a class="ref" href="card_1.html#T81">CARD_1:81</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T52">FINSEQ_3:52</a></i>
;<br/>


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