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<div class="add">

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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="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="E1:43"/><i><font color="Green" title="E24">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">S1</font>,<font color="Maroon" title="c2">S2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">A2</font>, <font color="Maroon" title="c6">B2</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S2</font>;<br/>

<b>set </b><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c5">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>;<br/>
<b>set </b><font color="Maroon" title="c8">B1</font> = <font color="Maroon" title="c6">B2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>;<br/>
<b>let </b><font color="Maroon" title="c9">h2</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c5">A2</font>,<font color="Maroon" title="c6">B2</font>;<br/><b>let </b><font color="Maroon" title="c10">h1</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:43"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c11">o1</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c12">o2</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S2</font>;<br/>



<b>assume </b><a NAME="E3:43"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c12">o2</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">o1</font>
 ;<br/>

<b>assume </b><a NAME="E4:43"/><i><font color="Green" title="E27">A4</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c5">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c6">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">x2</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="c12">o2</font>,<font color="Maroon" title="c5">A2</font>;<br/><b>let </b><font color="Maroon" title="c14">x1</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="c11">o1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span>;<br/>



<b>assume </b><a NAME="E5:43"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">f1</font> = <font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c16">f2</font> = <font color="Maroon" title="c13">x2</font>, <font color="Maroon" title="c17">g2</font> = <font color="Maroon" title="c9">h2</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c13">x2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="instalg1.html#E4:43"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msualg_6.html#T1" title="MSUALG_6:th.1">MSUALG_6:1</a></i>;<br/>
<a NAME="E7:43"/><i><font color="Green" title="E29">A6</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">A2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="instalg1.html#E3:43"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="instalg1.html#T25" target="_self" title="INSTALG1:th.25">Th25</a></i>;<br/>
<a NAME="E8:43"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f1</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="c11">o1</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">f2</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="c12">o2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="instalg1.html#E4:43"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c18">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="c15">f1</font>
 ;<br/><a NAME="E2:43_1"/>
 <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">S1</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/><b>then </b><font color="Maroon" title="c10">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="c11">o1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> = 
<font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1: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="c11">o1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E2:43"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1: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="c11">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="instalg1.html#E1:43_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c9">h2</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"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <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="c11">o1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="instalg1.html#E1:43_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">h2</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="c12">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="instalg1.html#E3:43"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c9">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="c12">o2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E5:43"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="instalg1.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="instalg1.html#E1:43_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon" title="c17">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h1</font> <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="c11">o1</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">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="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="instalg1.html#E4:43"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="instalg1.html#E5:43"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="instalg1.html#E1:43_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="msualg_3.html#T24" title="MSUALG_3:th.24">MSUALG_3:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:43"/>
<font color="Maroon" title="c10">h1</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E4:43"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="instalg1.html#E7:43"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="msualg_3.html#T24" title="MSUALG_3:th.24">MSUALG_3:24</a></i>;<br/>


</div>
