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<b>let </b><font color="Maroon">S</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="instalg1.html#V1" target="_self">feasible</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">T'</font> be    <a href="instalg1.html#M1" target="_self">Subsignature</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E2:28"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T</font> )
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">T</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span>
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green">E53</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a><br/>

<b>thus </b><a NAME="E6:28"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a><br/>

<b>thus </b><a NAME="E7:28"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">T</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">T</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">T</font> <a href="partfun1.html#K2">|</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="relat_1.html#T94">RELAT_1:94</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">T'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i/>
;<br/>

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E9:28"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 ;<br/>

<a NAME="E10:28"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font> &amp; the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T'</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T'</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">T'</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:28"/>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">T'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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