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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">U2</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U3</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U2</font>,<font color="Maroon">U3</font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font>
 ;<br/>

<b>thus </b><a NAME="E2:18"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font> implies for <font color="Olive">U1</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><br/> for <font color="Olive">h1</font>, <font color="Olive">h2</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Olive">U1</font>,<font color="Maroon">U2</font>  st <font color="Olive">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Olive">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h2</font> holds <br/><font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Olive">h2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
for <font color="Olive">U1</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><br/> for <font color="Olive">h1</font>, <font color="Olive">h2</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Olive">U1</font>,<font color="Maroon">U2</font>  st <font color="Olive">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Olive">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h2</font> holds <br/><font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Olive">h2</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:18"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">U1</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><br/> for <font color="Olive">h1</font>, <font color="Olive">h2</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Olive">U1</font>,<font color="Maroon">U2</font>  st <font color="Olive">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Olive">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h2</font> holds <br/><font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Olive">h2</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">F</font> <a href="msualg_3.html#R3">is_monomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E5:18"/>
not <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:18"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>
<b>set </b><font color="Maroon">C</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font>;<br/>
<a NAME="E8:18"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E9:18"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:18"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<a NAME="E12:18"/>
 ex <font color="Olive">U1</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ex <font color="Olive">h1</font>, <font color="Olive">h2</font> being   <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> st <br/>( <font color="Olive">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Olive">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">h2</font> &amp; <font color="Olive">h1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">h2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">U1</font> =  <a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>

<b>reconsider </b><font color="Maroon">FG</font> =  <a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> as    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">U1</font> ;<br/>
<a NAME="E2:18_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">FG</font> is <a href="msafree.html#V1">free</a>
 
<b>by </b><i><a class="ref" href="msafree.html#T17">MSAFREE:17</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>;<br/>
<a NAME="E3:18_2"/>
<font color="Maroon">FG</font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="ref" href="msafree.html#T15">MSAFREE:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FGj</font> = <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">Bj</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">FGj</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">x1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font> <b>by </b><i>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E6:18_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FGj</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">FGj</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">x2</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font> <b>by </b><i>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:18_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FGj</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:18_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">j</font>,<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E12:18_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font>, <font color="Maroon">h</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font>;<br/><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Olive">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:18_2_1"/>
<font color="Maroon">G</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:18_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:18_2_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_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="E1:18_2_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_1_1_1_1"/><b>then </b>
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_1_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_1_1_1_1_2"/><b>then </b>
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_1_1_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font>, <font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font>, <font color="Maroon">h</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,<font color="Maroon">Bj</font>;<br/><b>assume </b><a NAME="E1:18_2_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Olive">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:18_2_2"/>
<font color="Maroon">G</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">FG</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:18_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_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="E1:18_2_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_1_1"><b>case </b></a><a NAME="E1:18_2_2_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_2_1_1_1_1"/><b>then </b>
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:18_2_2_1_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_1_2"><b>case </b></a><a NAME="E1:18_2_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_2_1_1_1_2"/><b>then </b>
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:18_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:18_2_2_1_1_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D18">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="E3:18_2_2_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font>, <font color="Maroon">H</font> = <font color="Maroon">H</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">FG</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">h1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font><b> such that </b><br/><a NAME="E18:18_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="msafree.html#D5">MSAFREE:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">h2</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font><b> such that </b><br/><a NAME="E20:18_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font> <a href="pboole.html#R6">=</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="msafree.html#D5">MSAFREE:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">h1</font>
;<br/>

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

<b>thus </b><a NAME="E21:18_2"/>
( <font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> )
 <b>by </b><i>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_2_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_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="E1:18_2_3_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_3_1_1_1"><b>case </b></a><a NAME="E1:18_2_3_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_3_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:18_2_3_1_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E4:18_2_3_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>then reconsider </b><font color="Maroon">fg</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/><b>reconsider </b><font color="Maroon">fh</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FGj</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_2_3_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FGj</font>
 ;<br/><b>hence </b><font color="Maroon">fg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">fh</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E8:18_2_3_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/><a NAME="E9:18_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E10:18_2_3_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E11:18_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/><b>hence </b><a NAME="E12:18_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_3_1_1_2"><b>case </b></a><a NAME="E1:18_2_3_1_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:18_2_3_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E3:18_2_3_1_1_2"/><i><font color="Green">E90</font></i>: 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E91</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , </i>
;<br/><b>reconsider </b><font color="Maroon">h2'</font> = <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">f'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h2'</font>
<b>by </b><i>, , <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T2">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="E3:18_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E23:18_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<i><font color="Green">E93</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h1</font></span>)</span> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font> = 
<font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h2</font></span>)</span> <a href="msafree.html#K1">||</a> <font color="Maroon">FG</font>
<b>by </b><i>, </i>
;<br/>
<b>reconsider </b><font color="Maroon">Fh1</font> = <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h1</font>, <font color="Maroon">Fh2</font> = <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h2</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U3</font> ;<br/>
<a NAME="E26:18_2"/>
( <font color="Maroon">Fh1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font> &amp; <font color="Maroon">Fh2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font> )
 
<b>by </b><i>, , , <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>
<b>hence </b><a NAME="E27:18_2"/>
<font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h2</font>
 <b>by </b><i>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_5"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E1:18_2_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/><i><font color="Green">E96</font></i>: <font color="Maroon">g</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><i><font color="Green">E114</font></i>: <font color="Maroon">h</font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FGj</font>;<br/><b>assume </b><a NAME="E1:18_2_5_3"/>
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 ;<br/><a NAME="E2:18_2_5_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_2_5_3"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:18_2_5"/>
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E29:18_2"/>
<font color="Maroon">h1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:18"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
