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<b>let </b><font color="Maroon" title="c3">F2</font>, <font color="Maroon" title="c4">F3</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font> holds <br/><font color="Maroon" title="c3">F2</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) )
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green" title="E41">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F1</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F3</font> holds <br/><font color="Maroon" title="c4">F3</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) )
 ;<br/>

<a NAME="E3:47_1_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F3</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font> holds <br/><font color="Maroon" title="c3">F2</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F3</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green" title="E42">A4</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F3</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E1:47_1_2"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="prob_3.html#E2:47_1_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F3</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:47_1_2_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F2</font>
 ;<br/>

<b>hence </b><font color="Maroon" title="c3">F2</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F1</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="prob_3.html#E1:47_1_2"><i><font color="Green" title="E40">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F3</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="prob_3.html#E2:47_1_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="prob_3.html#E1:47_1_2_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="prob_3.html#E2:47_1_2_1"><i><font color="Green" title="E43">A5</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_2"/>
<font color="Maroon" title="c3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


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