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<b>set </b><font color="Maroon">SW</font> =  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Olive">v</font> );<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">i</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> ;<br/><a NAME="E3:2_1_1_1"/>
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">IIG</font> : <font color="Olive">v</font> is <a href="msualg_2.html#V1">with_const_op</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="msafree2.html#D1">MSAFREE2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E5:2_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">v</font> is <a href="msualg_2.html#V1">with_const_op</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">o</font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E7:2_1_1_1"/><i><font color="Green">E35</font></i>: 
( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D2">MSUALG_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:2_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> &amp;  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> :  ex <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">IIG</font> st <br/>( the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Olive">o</font>,<font color="Maroon">SCS</font></span>)</span> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="msualg_2.html#D4">MSUALG_2:def 4</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">SCS</font></span>)</span>;<br/><a NAME="E11:2_1_1_1"/>
not  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">SCS</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#T6">MSUALG_1:6</a></i>;<br/><a NAME="E12:2_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">SCS</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E13:2_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">SCS</font></span>)</span>
 ;<br/><a NAME="E14:2_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/> <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">SCS</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">IIG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <b>by </b><i>, </i>;<br/><a NAME="E17:2_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E19:2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E3:2_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

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

<b>assume </b><a NAME="E4:2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E5:2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K1">Constants</a> <font color="Maroon">SCS</font>,<font color="Maroon">x</font>
 ;<br/>


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