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<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="msualg_1.html#V1">strict</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>set </b><font color="Maroon">f1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">f2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">g1</font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">g2</font> = the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:39_1_1"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S</font>,<font color="Maroon">S1</font>
 <b>and </b><br/><a NAME="E2:39_1_1"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E28</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E4:39_1_1"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font>,the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S</font>,<font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E5:39_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:39_1_1"/><i><font color="Green">E32</font></i>: 
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_1"/><i><font color="Green">E33</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>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E1:39_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E1:39_1_1_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="algspec1.html#E5:39_1_1"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/><b>thus </b>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> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E5:39_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a>, </i>
<br/>.= 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="algspec1.html#T2" target="_self">Th2</a>, , <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:39_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S1</font> <a href="funct_2.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E6:39_1_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="algspec1.html#T3" target="_self">Th3</a>, <a class="ref" href="algspec1.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_2"/><i><font color="Green">E92</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>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="algspec1.html#E5:39_1_1"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/><b>thus </b>the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="txt" href="algspec1.html#E5:39_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E4:39_1_1"><i><font color="Green">E30</font></i></a>, , , <a class="txt" href="algspec1.html#E1:39_1_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<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> <span class="p2">(<span class="default">the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="algspec1.html#K1" target="_self">-indexing</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="algspec1.html#T2" target="_self">Th2</a>, <a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>
<br/>.= 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:39_1_1"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E6:39_1_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="algspec1.html#T3" target="_self">Th3</a>, <a class="ref" href="algspec1.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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