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

<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U2</font>, <font color="Olive" title="b2">U3</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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="b3">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U2</font>,<font color="Olive" title="b2">U3</font>  st <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U2</font>,<font color="Olive" title="b2">U3</font> holds <br/>( <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Olive" title="b1">U2</font>,<font color="Olive" title="b2">U3</font> iff for <font color="Olive" title="b4">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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="b5">h1</font>, <font color="Olive" title="b6">h2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b4">U1</font>,<font color="Olive" title="b1">U2</font>  st <font color="Olive" title="b5">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b4">U1</font>,<font color="Olive" title="b1">U2</font> &amp; <font color="Olive" title="b6">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b4">U1</font>,<font color="Olive" title="b1">U2</font> &amp; <font color="Olive" title="b3">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b5">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b6">h2</font> holds <br/><font color="Olive" title="b5">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b6">h2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">U2</font>, <font color="Maroon" title="c3">U3</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <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>  st <font color="Olive" title="b1">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> holds <br/>( <font color="Olive" title="b1">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> iff for <font color="Olive" title="b2">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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="b3">h1</font>, <font color="Olive" title="b4">h2</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b2">U1</font>,<font color="Maroon" title="c2">U2</font>  st <font color="Olive" title="b3">h1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b2">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b4">h2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b2">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Olive" title="b1">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b4">h2</font> holds <br/><font color="Olive" title="b3">h1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">h2</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies ( <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> iff for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> iff for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">C</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<span class="kw">thus </span><a NAME="E2:21"/>
( <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#V4" title="MSUALG_1:attr.4">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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> ) implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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></span><br/>

<a NAME="E2:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_1"/>
for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T17" target="_self" title="EXTENS_1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c7">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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
for <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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>
 <span class="kw">and </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:21"/>
not <font color="Maroon" title="c4">F</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:21"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">and </span><br/><a NAME="E8:21"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
not <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></span>;<br/>
<span class="kw">set </span><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="c8">j</font>;<br/>
<a NAME="E9:21"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c8">j</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c8">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E11:21"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E12:21"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E13:21"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<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="c8">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="c8">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 <span class="kw">and </span><br/><a NAME="E14:21"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<a NAME="E15:21"/>
 ex <font color="Olive" title="b1">U1</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">U1</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c12">U1</font> =  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c12">U1</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font> st <br/>( <font color="Olive" title="b1">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="Olive" title="b2">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="c4">F</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">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="b2">h2</font> &amp; <font color="Olive" title="b1">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">h2</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">FG</font> =  <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="E2:21_2"/>
<font color="Maroon" title="c13">FG</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T15" title="MSAFREE:th.15">MSAFREE:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">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="c8">j</font>, <font color="Maroon" title="c15">Bj</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c14">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="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c14">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="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">r</font> =  <a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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/>
<span class="kw">consider </span><font color="Maroon" title="c19">G</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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><span class="kw"> such that </span><br/><a NAME="E7:21_2"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <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="c8">j</font>,<font color="Maroon" title="c16">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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/>
<span class="kw">consider </span><font color="Maroon" title="c20">H</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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><span class="kw"> such that </span><br/><a NAME="E9:21_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">G</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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="b3">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="b3">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="b3">i</font></span>)</span> ) holds <br/><font color="Maroon" title="c21">G</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> </span><br/><span class="kw">let </span><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="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> ) implies <font color="Maroon" title="c21">G</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:21_2_1"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">G</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> </span><br/><span class="kw">thus </span><a NAME="E2:21_2_1"/>
<font color="Maroon" title="c21">G</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c21">G</font> or <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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<a NAME="E2:21_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_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="c8">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="c8">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:21_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="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/><div class="add"><a NAME="E2:21_2_1_1_1_1_1"/><span class="kw">then </span>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c24">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_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="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E2:21_2_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:21_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="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/><div class="add"><a NAME="E2:21_2_1_1_1_1_2"/><span class="kw">then </span>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c24">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_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="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E2:21_2_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:21_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="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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="b3">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="b3">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Olive" title="b1">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="b3">i</font></span>)</span> ) holds <br/><font color="Maroon" title="c23">G</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font></span><br/><span class="kw">let </span><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="c14">FGj</font>,<font color="Maroon" title="c15">Bj</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> ) implies <font color="Maroon" title="c23">G</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_2_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">G</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font></span><br/><span class="kw">thus </span><a NAME="E2:21_2_2"/>
<font color="Maroon" title="c23">G</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>)) )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2_2_1"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_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="c8">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="c8">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:21_2_2_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))</span><br/><div class="add"><a NAME="E2:21_2_2_1_1_1_1"/><span class="kw">then </span>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c26">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E1:21_2_2_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E1:21_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E19">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:21_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="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))</span><br/><div class="add"><a NAME="E2:21_2_2_1_1_1_2"/><span class="kw">then </span>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c26">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E3:21_2_2_1_1_1_2"/><span class="kw">then </span>
<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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E1:21_2_2_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_2_1"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:21_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K22</a>(<span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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></span>)</span>))
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c13">FG</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>;<br/>
<a NAME="E14:21_2"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
<font color="Maroon" title="c13">FG</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T17" title="MSAFREE:th.17">MSAFREE:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E16:21_2"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E17:21_2"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E19:21_2"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E20:21_2"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_2"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="ref" href="msafree.html#D5" title="MSAFREE:def.5">MSAFREE:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">h1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h2</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c12">U1</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U2</font> st <br/>( <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="Olive" title="b1">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="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="Olive" title="b1">h2</font> &amp; <font color="Maroon" title="c25">h1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">h2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c26">h2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> &amp; <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> )</span><br/>

<span class="kw">thus </span><a NAME="E21:21_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_2"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E19:21_2"><span class="lab"><font color="Green" title="E20">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> )</span><br/>

<span class="kw">reconsider </span><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="E23:21_2"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E16:21_2"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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="c29">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="c29">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_2_3"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c29">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="c29">i</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_2_3_1_1"/>
( <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c29">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:21_2_3_1_1_1"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c29">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="c29">i</font></span><br/><div class="add"><a NAME="E2:21_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">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="c8">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="c14">FGj</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">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="c8">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c14">FGj</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E2:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">FGj</font> implies <font color="Maroon" title="c30">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">fh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_2_3_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A26</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">FGj</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">fh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c30">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">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="c8">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="c32">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<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="c8">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E1:21_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<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="c8">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="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c31">fh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E26">A26</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:21_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A27</font></span>: 
<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="c8">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="c8">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/><a NAME="E7:21_2_3_1_1_1"/>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E8:21_2_3_1_1_1"/><span class="kw">then </span>
<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="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E17:21_2"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a></span>;<br/><a NAME="E9:21_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A28</font></span>: 
<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="c29">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="c8">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>;<br/><a NAME="E10:21_2_3_1_1_1"/>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c16">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E11:21_2_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">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="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E20:21_2"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:21_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="c29">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="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="txt" href="#E6:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E26">A27</font></span></a>, <a class="txt" href="#E9:21_2_3_1_1_1"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:21_2_3_1_1_2"/><span class="lab"><font color="Green" title="E24">A29</font></span>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c29">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="c29">i</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c30">f9</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">h29</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="c29">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c13">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="c29">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><a NAME="E4:21_2_3_1_1_2"/><span class="lab"><font color="Green" title="E25">A30</font></span>: 
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c16">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_2"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><a NAME="E5:21_2_3_1_1_2"/>
 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">IFEQ</a> <font color="Maroon" title="c29">i</font>,<font color="Maroon" title="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3_1_1_2"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>;<br/><span class="kw">then </span><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="c29">i</font> = 
<span class="p1">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E17:21_2"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a></span>
<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="c29">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E20:21_2"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E4:21_2_3_1_1_2"><span class="lab"><font color="Green" title="E25">A30</font></span></a></span>
;<br/><span class="kw">hence </span><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="c29">i</font> = 
<font color="Maroon" title="c30">f9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c31">h29</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<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="c29">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_2_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:21_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="c29">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="c29">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E25:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A31</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<a NAME="E26:21_2"/><span class="lab"><font color="Green" title="E24">A32</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E19:21_2"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></span>;<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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T8" target="_self" title="EXTENS_1:th.8">Th8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_2"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="extens_1.html#T8" target="_self" title="EXTENS_1:th.8">Th8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E28:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:21_2"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="txt" href="#E26:21_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="ref" href="extens_1.html#T18" target="_self" title="EXTENS_1:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c29">i</font> = <font color="Maroon" title="c8">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> )</span><br/><span class="kw">thus </span><a NAME="E1:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div><span class="lab"><font color="Green" title="E25">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="c14">FGj</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c17">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">h</font></span><br/><span class="kw">assume </span><a NAME="E1:21_2_5_1"/>
<font color="Maroon" title="c17">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_2_5_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_2_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E26">A34</font></span>: <font color="Maroon" title="c16">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="c8">j</font>,<font color="Maroon" title="c16">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E9:21_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>
;<br/><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="c8">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#K16" title="MSAFREE:func.16">Reverse</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_2_5"><span class="lab"><font color="Green" title="E26">A34</font></span></a>, <a class="txt" href="#E2:21_2_5"><span class="lab"><font color="Green" title="E25">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:21_2"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E20:21_2"><span class="lab"><font color="Green" title="E21">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:21"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
