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<b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>, <font color="Maroon">G</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green">E112</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>for <font color="Olive">x</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>  st <font color="Olive">x</font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Olive">y</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green">E113</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/>for <font color="Olive">x</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>  st <font color="Olive">x</font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Olive">y</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 ;<br/>

<a NAME="E3:29_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_2_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font><b> such that </b><br/><a NAME="E4:29_1_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="alg_1.html#R6" target="_self">is_representatives_FS</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:29_1_2_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">E</font>,<span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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