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<b>let </b><font color="Maroon">S1</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S3</font> be    <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> )
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green">E45</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">S1</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">S1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">f1</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">S2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span>
 <b>and </b><br/><a NAME="E5:62"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 <b>and </b><br/><a NAME="E6:62"/><i><font color="Green">E58</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S3</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S3</font> )
 <b>and </b><br/><a NAME="E7:62"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S3</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g2</font>
 <b>and </b><br/><a NAME="E8:62"/><i><font color="Green">E72</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">S2</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">S2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">f2</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">S3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" target="_self">PUA2MSS1:def 13</a><br/>

<b>set </b><font color="Maroon">f</font> = <font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>;<br/>
<b>thus </b><a NAME="E9:62"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, , , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" target="_self">PUA2MSS1:def 13</a><br/>

<a NAME="E10:62"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<b>hence </b><a NAME="E11:62"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S3</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S3</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> = 
<font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S3</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i><a class="txt" href="pua2mss1.html#E4:62"><i><font color="Green">E45</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></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="E13:62"/><i><font color="Green">E73</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">S1</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">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 ;<br/>

<a NAME="E14:62"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">f1</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">S2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> &amp; <font color="Maroon">g1</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">g1</font> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:62"/>
( <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T1" target="_self">Th1</a>, <a class="ref" href="pua2mss1.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E16:62"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <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">S3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="pua2mss1.html#E5:62"><i><font color="Green">E46</font></i></a>, </i>;<br/>


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