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<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">e</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E2:12_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:12_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> )
 <b>and </b><br/><a NAME="E5:12_1_1"/><i><font color="Green">E33</font></i>: 
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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:12_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">e</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E8:12_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2" target="_self">InputVertices</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E9:12_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E10:12_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MSA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , </i>;<br/>


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