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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">U1</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">U1</font>, <font color="Maroon">h2</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">U1</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>,<font color="Maroon">U1</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>,<font color="Maroon">U1</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<a NAME="E4:16"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>,<font color="Maroon">U1</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E5:16"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">s</font>,<font color="Maroon">X</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
;<br/>
<a NAME="E6:16"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>,<font color="Maroon">U1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Olive">s</font>,<font color="Maroon">X</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">FG</font> =  <a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:16_1"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>;<br/>
<a NAME="E3:16_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>] )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:16"/>
<font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font>
 <b>from </b><i><a class="ref" href="msafree1.html#S3">MSAFREE1:sch 3</a>(, , , );</i><br/>


</div>
