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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</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">A</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">U1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font>;<br/>









<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E28">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  ex <font color="Olive" title="b2">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>,<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c6">H</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">s'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s'</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<b>set </b><font color="Maroon" title="c7">SU</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c8">CA</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c9">SA</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">s'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c10">s'</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 <font color="Maroon" title="c2">S</font>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
<font color="Maroon" title="c10">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:37_1"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</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> 
 <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">s''</font> = <font color="Maroon" title="c10">s'</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_1"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><a NAME="E5:37_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E1:37_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="pralg_3.html#T27" target="_self" title="PRALG_3:th.27">Th27</a></i>;<br/><a NAME="E6:37_1"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/><a NAME="E7:37_1"/><b>then </b>
<font color="Maroon" title="c6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="pralg_3.html#E1:37_1"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">Hs</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:37_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c12">Hs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">Hs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">Hs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E10:37_1"/>
<font color="Maroon" title="c10">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:37_1"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
not <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><a NAME="E12:37_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">Hs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">Hs</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="pralg_3.html#E9:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:37_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s''</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</font>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E3:37_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:37_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">Hs</font> &amp; <font color="Maroon" title="c12">Hs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c16">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s''</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">i''</font> = <font color="Maroon" title="c16">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:37_1_1_1"/><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s''</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">F'</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">i''</font></span>)</span><b> such that </b><br/><a NAME="E5:37_1_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
( <font color="Maroon" title="c18">F'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i''</font> &amp; <font color="Maroon" title="c18">F'</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>,<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">i''</font> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E6:37_1_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F'</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s''</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="pralg_3.html#E6:37_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="pralg_3.html#E5:37_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="pralg_3.html#T28" target="_self" title="PRALG_3:th.28">Th28</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E8:37_1_1_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <font color="Maroon" title="c19">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">i''</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s''</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c19">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s''</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/><a NAME="E9:37_1_1_1"/><i><font color="Green" title="E42">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F'</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s''</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">i''</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s''</font>
 ;<br/><a NAME="E10:37_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F'</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">s''</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">Hs</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E9:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:37_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E12:37_1_1_1"/>
<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s''</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="pralg_3.html#E8:37_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_1_1_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:37_1_1"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s''</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E4:37_1_1"><i><font color="Green" title="E36">A9</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="E9:37_1_1"/>
<font color="Maroon" title="c13">x</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 <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:37_1"/>
<font color="Maroon" title="c6">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font> is   <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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E9:37_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="pralg_3.html#E11:37_1"><i><font color="Green" title="E33">A6</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="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c6">H</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">H</font>
;<br/>

<a NAME="E6:37"/><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:37_2"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</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="c12">x</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">o</font>,<font color="Maroon" title="c4">U1</font>;<br/>

<b>set </b><font color="Maroon" title="c13">s'</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font>;<br/>
<a NAME="E2:37_2"/><i><font color="Green" title="E31">A17</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">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="E3:37_2"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span>
 
;<br/>
<b>thus </b><a NAME="E4:37_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1"/><i><font color="Green" title="E33">A19</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c14">f</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span>;<br/><a NAME="E2:37_2_1_1_1"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="pralg_2.html#T11" title="PRALG_2:th.11">PRALG_2:11</a></i>;<br/><a NAME="E3:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="pralg_3.html#T12" target="_self" title="PRALG_3:th.12">Th12</a></i>;<br/><a NAME="E5:37_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_2_1_1_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="pralg_3.html#T29" target="_self" title="PRALG_3:th.29">Th29</a></i>;<br/><b>then </b><i><font color="Green" title="E36">A22</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E7:37_2_1_1_1"/>
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c11">o</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i1</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="pralg_3.html#T9" target="_self" title="PRALG_3:th.9">Th9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">Co</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:37_2_1_1_1"/><i><font color="Green" title="E37">A23</font></i>: 
( <font color="Maroon" title="c15">Co</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">Co</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">Co</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c11">o</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i1</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_2"/>
<font color="Maroon" title="c16">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">ii</font> = <font color="Maroon" title="c16">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><b>consider </b><font color="Maroon" title="c18">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">ii</font></span>)</span><b> such that </b><br/><a NAME="E4:37_2_1_1_1_2"/><i><font color="Green" title="E38">A24</font></i>: 
( <font color="Maroon" title="c18">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">ii</font> &amp; <font color="Maroon" title="c18">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>,<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">ii</font> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E5:37_2_1_1_1_2"/><i><font color="Green" title="E39">A25</font></i>: 
<font color="Maroon" title="c18">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_2_1_1_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="pralg_3.html#T12" target="_self" title="PRALG_3:th.12">Th12</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E2:37_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_2_1_1_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="pralg_3.html#E4:37_2_1_1_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="pralg_3.html#T28" target="_self" title="PRALG_3:th.28">Th28</a></i>
<br/>.= 
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">ii</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E4:37_2_1_1_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="pralg_3.html#E5:37_2_1_1_1_2"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i'</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_1"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="pralg_3.html#T10" target="_self" title="PRALG_3:th.10">Th10</a></i>
;<br/></div><b>end;</b></div><a NAME="E11:37_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37_2_1_1_1"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_2_1_1_1"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E12:37_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_2_1_1_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="pralg_3.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_2"/><i><font color="Green" title="E33">A26</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</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/><div class="add"><b>set </b><font color="Maroon" title="c14">f</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>;<br/><i><font color="Green" title="E34">A27</font></i>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K13" title="PRALG_2:func.13">OPS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c11">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</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">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K1" title="PRALG_2:func.1">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D20" title="PRALG_2:def.20">PRALG_2:def 20</a></i>
<br/>.= 
 <a href="pralg_2.html#K1" title="PRALG_2:func.1">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><div><i><font color="Green" title="E35">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">y</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">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>;<br/><a NAME="E1:37_2_1_1_2_2"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K1" title="PRALG_2:func.1">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="pralg_3.html#T19" target="_self" title="PRALG_3:th.19">Th19</a></i>;<br/><a NAME="E2:37_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E36">A29</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c15">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E2:37_2_1_1_2"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="pralg_2.html#D6" title="PRALG_2:def.6">PRALG_2:def 6</a></i>;<br/><a NAME="E3:37_2_1_1_2_2"/>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="pralg_3.html#T18" target="_self" title="PRALG_3:th.18">Th18</a></i>;<br/><a NAME="E4:37_2_1_1_2_2"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:37_2_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E2:37_2_1_1_2_2"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">f1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</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="pralg_2.html#T15" title="PRALG_2:th.15">PRALG_2:15</a></i>;<br/><b>set </b><font color="Maroon" title="c16">D</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">ss</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> ;<br/><a NAME="E5:37_2_1_1_2"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c11">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">ss</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T15" target="_self" title="PRALG_3:th.15">Th15</a></i>;<br/><a NAME="E6:37_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E2:37_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="pralg_3.html#T29" target="_self" title="PRALG_3:th.29">Th29</a></i>;<br/><b>then </b><i><font color="Green" title="E37">A31</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E8:37_2_1_1_2"/>
<font color="Maroon" title="c15">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a></i>;<br/><a NAME="E9:37_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A32</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pralg_2.html#T16" title="PRALG_2:th.16">PRALG_2:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">i'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_2_4"/>
<font color="Maroon" title="c17">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c18">ii</font> = <font color="Maroon" title="c17">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><b>consider </b><font color="Maroon" title="c19">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">ii</font></span>)</span><b> such that </b><br/><a NAME="E4:37_2_1_1_2_4"/><i><font color="Green" title="E39">A33</font></i>: 
( <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>,<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">ii</font> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/><a NAME="E5:37_2_1_1_2_4"/><i><font color="Green" title="E40">A34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">ii</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E4:37_2_1_1_2_4"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>;<br/><a NAME="E6:37_2_1_1_2_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">F1</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="c2">S</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:37_2_1_1_2_4"/><b>then </b><i><font color="Green" title="E41">A35</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">F1</font>
 ;<br/><a NAME="E8:37_2_1_1_2_4"/><i><font color="Green" title="E42">A36</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">F1</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">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T13" target="_self" title="PRALG_3:th.13">Th13</a></i>;<br/><i><font color="Green" title="E43">A37</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#D7" title="MSUALG_3:def.7">MSUALG_3:def 7</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">F1</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">o</font></span>)</span></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c12">x</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E8:37_2_1_1_2_4"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="pralg_2.html#D8" title="PRALG_2:def.8">PRALG_2:def 8</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c11">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></i>
<br/>.= 
 <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">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E7:37_2_1_1_2_4"><i><font color="Green" title="E41">A35</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
;<br/><a NAME="E10:37_2_1_1_2_4"/>
 <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">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><a NAME="E11:37_2_1_1_2_4"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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">o</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">ss</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E5:37_2_1_1_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">ff</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:37_2_1_1_2_4"/><i><font color="Green" title="E44">A38</font></i>: 
( <font color="Maroon" title="c20">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c11">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">ss</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E14:37_2_1_1_2_4"/>
<font color="Maroon" title="c20">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">ff</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E13:37_2_1_1_2_4"><i><font color="Green" title="E44">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E16:37_2_1_1_2_4"/><i><font color="Green" title="E45">A39</font></i>: 
( <font color="Maroon" title="c21">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">gg</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">o</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">ss</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="pralg_3.html#E13:37_2_1_1_2_4"><i><font color="Green" title="E44">A38</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_2_4_2"/><i><font color="Green" title="E46">A40</font></i>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">o</font></span>)</span>
 ;<br/><a NAME="E2:37_2_1_1_2_4_2"/><b>then </b>
<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">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">s1</font> = <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">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E4:37_2_1_1_2_4_2"/>
<font color="Maroon" title="c12">x</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="c11">o</font>,<font color="Maroon" title="c4">U1</font>
 ;<br/><a NAME="E5:37_2_1_1_2_4_2"/><b>then </b><i><font color="Green" title="E47">A41</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c11">o</font></span>)</span></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/><b>then </b><i><font color="Green" title="E48">A42</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">x</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c11">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <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">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="E7:37_2_1_1_2_4_2"/><i><font color="Green" title="E49">A43</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c11">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E5:37_2_1_1_2_4_2"><i><font color="Green" title="E47">A41</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E8:37_2_1_1_2_4_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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="c11">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="txt" href="pralg_3.html#E5:37_2_1_1_2_4_2"><i><font color="Green" title="E47">A41</font></i></a>, <a class="txt" href="pralg_3.html#E6:37_2_1_1_2_4_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E9:37_2_1_1_2_4_2"/><b>then </b><i><font color="Green" title="E50">A44</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="txt" href="pralg_3.html#E6:37_2_1_1_2_4_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="txt" href="pralg_3.html#E7:37_2_1_1_2_4_2"><i><font color="Green" title="E49">A43</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><i><font color="Green" title="E51">A45</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</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">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</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="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="pralg_3.html#T14" target="_self" title="PRALG_3:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c23">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><i><font color="Green" title="E52">A46</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</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">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</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="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="pralg_3.html#T14" target="_self" title="PRALG_3:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c23">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</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="c12">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c24">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> = 
<font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E5:37_2_1_1_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="pralg_3.html#E1:37_2_1_1_2_4_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E4:37_2_1_1_2_4"><i><font color="Green" title="E39">A33</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_2_1_1_2_4_2"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="pralg_3.html#E10:37_2_1_1_2_4_2"><i><font color="Green" title="E51">A45</font></i></a>, <a class="txt" href="pralg_3.html#E11:37_2_1_1_2_4_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="pralg_3.html#T28" target="_self" title="PRALG_3:th.28">Th28</a></i>
;<br/></div><b>end;</b></div><a NAME="E18:37_2_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E9:37_2_1_1_2_4"><i><font color="Green" title="E43">A37</font></i></a>, <a class="txt" href="pralg_3.html#E16:37_2_1_1_2_4"><i><font color="Green" title="E45">A39</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i'</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">ii</font></span>)</span></span>)</span> <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="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">ii</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E2:37_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="pralg_3.html#E4:37_2_1_1_2_4"><i><font color="Green" title="E39">A33</font></i></a>, <a class="txt" href="pralg_3.html#E5:37_2_1_1_2_4"><i><font color="Green" title="E40">A34</font></i></a>, <a class="ref" href="pralg_3.html#T28" target="_self" title="PRALG_3:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i'</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="pralg_3.html#T23" target="_self" title="PRALG_3:th.23">Th23</a></i>
;<br/><b>hence </b><a NAME="E20:37_2_1_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i'</font>
 ;<br/></div><b>end;</b></div><a NAME="E11:37_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f1</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E7:37_2_1_1_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_2_1_1_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E12:37_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:37"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/>

<b>consider </b><font color="Maroon" title="c12">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c4">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c11">i</font></span>)</span><b> such that </b><br/><a NAME="E2:37_3"/><i><font color="Green" title="E31">A47</font></i>: 
( <font color="Maroon" title="c12">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c12">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>,<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<i><font color="Green" title="E32">A48</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">H</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">H</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>

;<br/>
<a NAME="E4:37_3"/><i><font color="Green" title="E33">A49</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F1</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="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">s'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_2"/>
<font color="Maroon" title="c13">s'</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 <font color="Maroon" title="c2">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">s1</font> = <font color="Maroon" title="c13">s'</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E3:37_3_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="pralg_3.html#T27" target="_self" title="PRALG_3:th.27">Th27</a></i>;<br/><a NAME="E4:37_3_2"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">f'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:37_3_2"/><i><font color="Green" title="E34">A50</font></i>: 
( <font color="Maroon" title="c15">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>, <font color="Olive" title="b2">s1</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> : verum </span>}</span> </span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E7:37_3_2"/><i><font color="Green" title="E35">A51</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_3_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_3_2_1"/><i><font color="Green" title="E36">A52</font></i>: 
( <font color="Maroon" title="c17">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f'</font> &amp; <font color="Maroon" title="c15">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:37_3_2_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E6:37_3_2"><i><font color="Green" title="E34">A50</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_3_2_1"><i><font color="Green" title="E36">A52</font></i></a>, <a class="ref" href="pralg_3.html#T29" target="_self" title="PRALG_3:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E5:37_3_2_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37_3_2"><i><font color="Green" title="E34">A50</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_3_2_1"><i><font color="Green" title="E36">A52</font></i></a>, <a class="ref" href="pralg_3.html#D2" target="_self" title="PRALG_3:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E36">A53</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E3:37_3"><i><font color="Green" title="E32">A48</font></i></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3" target="_self" title="PRALG_3:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a></i>
;<br/><a NAME="E9:37_3_2"/><b>then </b><i><font color="Green" title="E37">A54</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37_3_2"><i><font color="Green" title="E34">A50</font></i></a>, <a class="txt" href="pralg_3.html#E7:37_3_2"><i><font color="Green" title="E35">A51</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><i><font color="Green" title="E38">A55</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span>

<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font>
<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="37_3_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_2_4"/><i><font color="Green" title="E39">A56</font></i>: 
<font color="Maroon" title="c16">x</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="c4">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font>
 ;<br/><a NAME="E2:37_3_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="pralg_3.html#T29" target="_self" title="PRALG_3:th.29">Th29</a></i>;<br/><a NAME="E3:37_3_2_4"/><b>then </b><i><font color="Green" title="E40">A57</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#D2" target="_self" title="PRALG_3:def.2">Def2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c14">s1</font></span>)</span>,<font color="Maroon" title="c11">i</font></span>)</span> <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="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E8:37_3_2"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="pralg_3.html#E9:37_3_2"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="pralg_3.html#E1:37_3_2_4"><i><font color="Green" title="E39">A56</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</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="pralg_3.html#E3:37_3_2_4"><i><font color="Green" title="E40">A57</font></i></a>, <a class="ref" href="pralg_3.html#D2" target="_self" title="PRALG_3:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="pralg_3.html#E2:37_3"><i><font color="Green" title="E31">A47</font></i></a>, <a class="txt" href="pralg_3.html#E1:37_3_2_4"><i><font color="Green" title="E39">A56</font></i></a>, <a class="ref" href="pralg_3.html#T28" target="_self" title="PRALG_3:th.28">Th28</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:37_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s'</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E9:37_3_2"><i><font color="Green" title="E37">A54</font></i></a>, <a class="txt" href="pralg_3.html#E10:37_3_2"><i><font color="Green" title="E38">A55</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_3"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</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="pralg_3.html#E2:37_3"><i><font color="Green" title="E31">A47</font></i></a>, <a class="txt" href="pralg_3.html#E3:37_3"><i><font color="Green" title="E32">A48</font></i></a>, <a class="txt" href="pralg_3.html#E4:37_3"><i><font color="Green" title="E33">A49</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:37"/>
( <font color="Maroon" title="c10">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c4">U1</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="pralg_3.html#E6:37"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/>


</div>
