<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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="c2">U2</font>, <font color="Maroon" title="c3">U3</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="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>
 ;<br/>

<b>thus </b><a NAME="E2:17"/>
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font> implies for <font color="Olive" title="b1">U1</font> being <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="c1">S</font><br/> for <font color="Olive" title="b2">h1</font>, <font color="Olive" title="b3">h2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b2">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b3">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">h2</font> holds <br/><font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">h2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:17_1"/>
for <font color="Olive" title="b1">U1</font> being <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="c1">S</font><br/> for <font color="Olive" title="b2">h1</font>, <font color="Olive" title="b3">h2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b2">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b3">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">h2</font> holds <br/><font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">h2</font>
 <b>by </b><i><a class="ref" href="extens_1.html#T17" target="_self" title="EXTENS_1:th.17">Th17</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">U1</font> being <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="c1">S</font><br/> for <font color="Olive" title="b2">h1</font>, <font color="Olive" title="b3">h2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b2">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b3">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">h2</font> holds <br/><font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">h2</font>
 <b>and </b><br/><a NAME="E4:17"/><i><font color="Green" title="E8">A3</font></i>: 
not <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c3">U3</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E5:17"/>
not <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 
<b>by </b><i><a class="txt" href="extens_1.html#E1:17"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="extens_1.html#E4:17"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:17"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c6">j</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="c1">S</font> &amp; not <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">B</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">C</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font>;<br/>
<a NAME="E8:17"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</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="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>;<br/>
<a NAME="E9:17"/>
( the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</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="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:17"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x2</font> )
 <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E8:17"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>
<a NAME="E12:17"/>
 ex <font color="Olive" title="b1">U1</font> being <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="c1">S</font> ex <font color="Olive" title="b2">h1</font>, <font color="Olive" title="b3">h2</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U1</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> st <br/>( <font color="Olive" title="b2">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b3">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b2">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">h2</font> &amp; <font color="Olive" title="b2">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">h2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c12">U1</font> =  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">FG</font> =  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c12">U1</font> ;<br/>
<a NAME="E2:17_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c13">FG</font> is <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>
 
<b>by </b><i><a class="ref" href="msafree.html#T17" title="MSAFREE:th.17">MSAFREE:17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">r</font> =  <a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<a NAME="E3:17_2"/>
<font color="Maroon" title="c13">FG</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="msafree.html#T15" title="MSAFREE:th.15">MSAFREE:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">FGj</font> = <font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c16">Bj</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> as  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>  <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c15">FGj</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c10">x1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,<font color="Maroon" title="c16">Bj</font> <b>by </b><i><a class="txt" href="extens_1.html#E11:17"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c15">FGj</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c11">x2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,<font color="Maroon" title="c16">Bj</font> <b>by </b><i><a class="txt" href="extens_1.html#E11:17"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> $1,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c17">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</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="c19">G</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="c1">S</font><b> such that </b><br/><a NAME="E8:17_2"/><i><font color="Green" title="E13">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c1">S</font> holds <br/><font color="Maroon" title="c19">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> $1,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c18">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</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="c20">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="c1">S</font><b> such that </b><br/><a NAME="E10:17_2"/><i><font color="Green" title="E14">A9</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c1">S</font> holds <br/><font color="Maroon" title="c20">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">G</font> be    <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="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c22">g</font>, <font color="Maroon" title="c23">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,<font color="Maroon" title="c16">Bj</font>;<br/><b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green" title="E15">A10</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c1">S</font> holds <br/><font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:17_2_1"/>
<font color="Maroon" title="c21">G</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">i</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="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:17_2_1_1"/>
<font color="Maroon" title="c24">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> <font color="Maroon" title="c21">G</font>
 ;<br/>

<a NAME="E2:17_2_1_1"/><b>then </b><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c24">i</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="c1">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="17_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1_1_1"/>
( <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c24">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:17_2_1_1_1_1_1"/>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_1_1_1_1_1"/><b>then </b>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c24">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_1_1_1"/>
<font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="extens_1.html#E2:17_2_1_1"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:17_2_1_1_1_1_2"/>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_1_1_1_1_2"/><b>then </b>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c24">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:17_2_1_1_1_1_2"/>
<font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="extens_1.html#E2:17_2_1_1"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_2_1_1"/>
<font color="Maroon" title="c21">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c21">G</font> = <font color="Maroon" title="c19">G</font>, <font color="Maroon" title="c22">H</font> = <font color="Maroon" title="c20">H</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="extens_1.html#E8:17_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="extens_1.html#E10:17_2"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">G</font> be   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c24">g</font>, <font color="Maroon" title="c25">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,<font color="Maroon" title="c16">Bj</font>;<br/><b>assume </b><a NAME="E1:17_2_2"/><i><font color="Green" title="E15">A12</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c1">S</font> holds <br/><font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c24">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:17_2_2"/>
<font color="Maroon" title="c23">G</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">i</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="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:17_2_2_1"/><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c26">i</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="c1">S</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_2_1_1_1"/>
( <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c26">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_2_1_1_1_1"><b>case </b></a><a NAME="E1:17_2_2_1_1_1_1"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_2_1_1_1_1"/><b>then </b>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c26">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c24">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:17_2_2_1_1_1_1"/>
<font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_2_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_2_1_1_1_1"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_2_1_1_1_2"><b>case </b></a><a NAME="E1:17_2_2_1_1_1_2"/>
<font color="Maroon" title="c26">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_2_1_1_1_2"/><b>then </b>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c26">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c24">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E3:17_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_2"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_2_1"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E4:17_2_2_1_1_1_2"/>
<font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_2_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2_2_1"/>
<font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c23">G</font> = <font color="Maroon" title="c21">G</font>, <font color="Maroon" title="c24">H</font> = <font color="Maroon" title="c22">H</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <b>by </b><i><a class="txt" href="extens_1.html#E8:17_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="extens_1.html#E10:17_2"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">h1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font><b> such that </b><br/><a NAME="E16:17_2"/><i><font color="Green" title="E15">A15</font></i>: 
( <font color="Maroon" title="c25">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">G</font> )
 <b>by </b><i><a class="txt" href="extens_1.html#E2:17_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">h2</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font><b> such that </b><br/><a NAME="E18:17_2"/><i><font color="Green" title="E16">A16</font></i>: 
( <font color="Maroon" title="c26">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">H</font> )
 <b>by </b><i><a class="txt" href="extens_1.html#E2:17_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">h1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c26">h2</font>
;<br/>

<b>thus </b><a NAME="E19:17_2"/>
( <font color="Maroon" title="c25">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c26">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c2">U2</font> )
 <b>by </b><i><a class="txt" href="extens_1.html#E16:17_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="extens_1.html#E18:17_2"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_2_3"/><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c27">i</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="c1">S</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_3_1_1"/>
( <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c27">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_3_1_1_1"><b>case </b></a><a NAME="E1:17_2_3_1_1_1"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E19">A19</font></i>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c27">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c17">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E3:17_2_3_1_1_1"/><i><font color="Green" title="E20">A20</font></i>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c27">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c18">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E4:17_2_3_1_1_1"/><i><font color="Green" title="E21">A21</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</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="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">fg</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">fh</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">FGj</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="extens_1.html#E4:17_2_3_1_1_1"><i><font color="Green" title="E21">A21</font></i></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:17_2_3_1_1_1_1"/><i><font color="Green" title="E22">A22</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">FGj</font>
 ;<br/><b>hence </b><font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</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="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
<b>by </b><i><a class="txt" href="extens_1.html#E11:17"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3_1_1_1_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</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="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
<b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_1_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c29">fh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font>
<b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_1_1"><i><font color="Green" title="E22">A22</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E8:17_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E22">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/><a NAME="E9:17_2_3_1_1_1"/><i><font color="Green" title="E23">A24</font></i>: 
<font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E10:17_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="extens_1.html#E18:17_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="extens_1.html#E3:17_2_3_1_1_1"><i><font color="Green" title="E20">A20</font></i></a></i>;<br/><a NAME="E10:17_2_3_1_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E8:17_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="extens_1.html#E16:17_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="extens_1.html#E2:17_2_3_1_1_1"><i><font color="Green" title="E19">A19</font></i></a></i>;<br/><a NAME="E11:17_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/><b>hence </b><a NAME="E12:17_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3_1_1_1"><i><font color="Green" title="E18">A18</font></i></a>, <a class="txt" href="extens_1.html#E8:17_2_3_1_1_1"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="extens_1.html#E9:17_2_3_1_1_1"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_3_1_1_2"><b>case </b></a><a NAME="E1:17_2_3_1_1_2"/><i><font color="Green" title="E18">A25</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font>
 ;<br/><div class="add"><a NAME="E2:17_2_3_1_1_2"/><b>then </b><i><font color="Green" title="E19">A26</font></i>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c27">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c17">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><a NAME="E3:17_2_3_1_1_2"/><i><font color="Green" title="E20">A27</font></i>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c27">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c18">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3_1_1_2"><i><font color="Green" title="E18">A25</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>;<br/><i><font color="Green" title="E21">A28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> = 
<span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
<b>by </b><i><a class="txt" href="extens_1.html#E8:17_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="extens_1.html#E16:17_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="extens_1.html#E2:17_2_3_1_1_2"><i><font color="Green" title="E19">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
<b>by </b><i><a class="txt" href="extens_1.html#E10:17_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="extens_1.html#E18:17_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="extens_1.html#E3:17_2_3_1_1_2"><i><font color="Green" title="E20">A27</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c28">h2'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">f'</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> = 
<font color="Maroon" title="c29">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">h2'</font>
<b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="extens_1.html#E4:17_2_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
<b>by </b><i><a class="txt" href="extens_1.html#E1:17_2_3"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E21:17_2"/><b>then </b><i><font color="Green" title="E17">A29</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<i><font color="Green" title="E18">A30</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c25">h1</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font> = 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font></span>)</span>
<b>by </b><i><a class="ref" href="extens_1.html#T8" target="_self" title="EXTENS_1:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c26">h2</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c13">FG</font>
<b>by </b><i><a class="txt" href="extens_1.html#E21:17_2"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="extens_1.html#T8" target="_self" title="EXTENS_1:th.8">Th8</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c27">Fh1</font> = <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c25">h1</font>, <font color="Maroon" title="c28">Fh2</font> = <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c26">h2</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c3">U3</font> ;<br/>
<a NAME="E24:17_2"/>
( <font color="Maroon" title="c27">Fh1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c3">U3</font> &amp; <font color="Maroon" title="c28">Fh2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">U1</font>,<font color="Maroon" title="c3">U3</font> )
 
<b>by </b><i><a class="txt" href="extens_1.html#E1:17"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="extens_1.html#E16:17_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="extens_1.html#E18:17_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></i>;<br/>
<b>hence </b><a NAME="E25:17_2"/>
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c25">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c26">h2</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E22:17_2"><i><font color="Green" title="E18">A30</font></i></a>, <a class="ref" href="extens_1.html#T18" target="_self" title="EXTENS_1:th.18">Th18</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_5"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c29">i</font> = <font color="Maroon" title="c6">j</font>;<br/><b>thus </b><a NAME="E1:17_2_5"/>
<font color="Maroon" title="c29">i</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="c1">S</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/><i><font color="Green" title="E19">A31</font></i>: <font color="Maroon" title="c17">g</font> = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c17">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c23">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E8:17_2"><i><font color="Green" title="E13">A8</font></i></a></i>
;<br/><i><font color="Green" title="E20">A32</font></i>: <font color="Maroon" title="c18">h</font> = 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c18">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c24">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<b>by </b><i><a class="txt" href="extens_1.html#E7:17"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="extens_1.html#E10:17_2"><i><font color="Green" title="E14">A9</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">FGj</font>;<br/><b>assume </b><a NAME="E1:17_2_5_3"/>
<font color="Maroon" title="c17">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">h</font>
 ;<br/><a NAME="E2:17_2_5_3"/><b>then </b>
<font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:17_2_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="extens_1.html#E11:17"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:17_2_5"/>
<font color="Maroon" title="c23">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">H</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E2:17_2_5"><i><font color="Green" title="E19">A31</font></i></a>, <a class="txt" href="extens_1.html#E3:17_2_5"><i><font color="Green" title="E20">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:17_2"/>
<font color="Maroon" title="c25">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">h2</font>
 <b>by </b><i><a class="txt" href="extens_1.html#E16:17_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="extens_1.html#E18:17_2"><i><font color="Green" title="E16">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:17"/>
contradiction
 <b>by </b><i><a class="txt" href="extens_1.html#E3:17"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>


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