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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font>;<br/>





<b>set </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h2</font> <a href="msualg_6.html#K1" title="MSUALG_6:func.1">**</a> <font color="Maroon" title="c3">h1</font>;<br/>
<b>let </b><font color="Maroon" title="c6">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c3">h1</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c4">h2</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c7">a</font></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="msualg_6.html#T1" target="_self" title="MSUALG_6:th.1">Th1</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="msualg_6.html#T2" target="_self" title="MSUALG_6:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c11">i</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="c8">f</font>
 ;<br/><a NAME="E2:11_1"/><b>then </b><i><font color="Green" title="E11">A4</font></i>: 
<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="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f1</font> = <font color="Maroon" title="c3">h1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>, <font color="Maroon" title="c13">f2</font> = <font color="Maroon" title="c4">h2</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> ;<br/><a NAME="E4:11_1"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="msualg_6.html#K1" title="MSUALG_6:func.1">**</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f1</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/><a NAME="E5:11_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="msualg_6.html#E1:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="msualg_3.html#T24" title="MSUALG_3:th.24">MSUALG_3:24</a></i>;<br/><a NAME="E6:11_1"/>
( <font color="Maroon" title="c3">h1</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">o</font></span>)</span> )
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="msualg_6.html#T2" target="_self" title="MSUALG_6:th.2">Th2</a></i>;<br/><a NAME="E7:11_1"/><b>then </b><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c13">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="msualg_6.html#E1:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="msualg_3.html#T24" title="MSUALG_3:th.24">MSUALG_3:24</a></i>;<br/><a NAME="E8:11_1"/><i><font color="Green" title="E15">A8</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c6">o</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/><a NAME="E9:11_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="msualg_6.html#E1:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E10:11_1"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="msualg_6.html#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="msualg_6.html#E1:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="msualg_6.html#E2:11_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="msualg_6.html#E8:11_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E11:11_1"/>
<font color="Maroon" title="c10">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="msualg_6.html#K1" title="MSUALG_6:func.1">**</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E4:11_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="msualg_6.html#E5:11_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="msualg_6.html#E7:11_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon" title="c4">h2</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="msualg_6.html#K1" title="MSUALG_6:func.1">**</a> <font color="Maroon" title="c3">h1</font></span>)</span> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="msualg_6.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="msualg_3.html#T24" title="MSUALG_3:th.24">MSUALG_3:24</a></i>;<br/>


</div>
