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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">S'</font> be  non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>







<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S'</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">R</font> = <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>;<br/>
<b>set </b><font color="Maroon" title="c6">F</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">G</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">f'</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c9">g'</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font>;<br/>
<a NAME="E2:44"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="algspec1.html#R1" title="ALGSPEC1:pred.1">form_a_replacement_in</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="algspec1.html#T32" target="_self" title="ALGSPEC1:th.32">Th32</a></i>;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green" title="E38">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="algspec1.html#E2:44"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></i>;<br/>
<a NAME="E5:44"/><i><font color="Green" title="E39">A5</font></i>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="algspec1.html#E2:44"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="algspec1.html#D4" target="_self" title="ALGSPEC1:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E6:44"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a>,<a class="ref" href="instalg1.html#D2" title="INSTALG1:def.2">INSTALG1:def 2</a><br/>

<a NAME="E7:44"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> &amp; the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> <a href="algspec1.html#K1" title="ALGSPEC1:func.1">-indexing</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> )
 
<b>by </b><i><a class="ref" href="algspec1.html#T10" target="_self" title="ALGSPEC1:th.10">Th10</a></i>;<br/>
<a NAME="E9:44"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E10:44"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S'</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="algspec1.html#E4:44"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="algspec1.html#E5:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>

<a NAME="E11:44"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S'</font> )
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E12:44"/><b>then </b><i><font color="Green" title="E41">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="algspec1.html#E5:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E13:44"/><i><font color="Green" title="E42">A8</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E43">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:44_1"/><i><font color="Green" title="E44">A10</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E5:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="algspec1.html#E3:44_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/><a NAME="E5:44_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/><a NAME="E6:44_1"/><b>then </b><i><font color="Green" title="E45">A11</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E3:44_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E7:44_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E3:44"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/><a NAME="E8:44_1"/><b>then </b>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E3:44_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E9:44_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="algspec1.html#E1:44_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="algspec1.html#E6:44_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:44"/><b>then </b><i><font color="Green" title="E43">A12</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="algspec1.html#E12:44"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="algspec1.html#E13:44"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> = 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E13:44"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E15:44"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c10">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c11">p</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E17:44"/><i><font color="Green" title="E44">A13</font></i>: 
( <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:44"/><i><font color="Green" title="E45">A14</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">a</font> )
 <b>by </b><i><a class="txt" href="algspec1.html#E5:44"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E20:44"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="txt" href="algspec1.html#E17:44"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E21:44"/><b>then </b>
<font color="Maroon" title="c11">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E22:44"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">a</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="algspec1.html#E19:44"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E24:44"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> )
 
<b>by </b><i><a class="txt" href="algspec1.html#E1:44"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="algspec1.html#E3:44"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="algspec1.html#E8:44"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="algspec1.html#E17:44"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="algspec1.html#E19:44"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">p</font> = 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
<b>by </b><i><a class="txt" href="algspec1.html#E22:44"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
<br/>.= 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="algspec1.html#K2" title="ALGSPEC1:func.2">with-replacement</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="algspec1.html#E17:44"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><br/>


</div>
