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

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E1:52_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="instalg1.html#E1:52"><i><font color="Green">E28</font></i></a>, <a class="ref" href="instalg1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/><a NAME="E4:52_1_1_1"/>
 <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/><a NAME="E5:52_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/><a NAME="E6:52_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #) &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="msafree.html#D13">MSAFREE:def 13</a>, <a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>;<br/><a NAME="E7:52_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="instalg1.html#E1:52"><i><font color="Green">E28</font></i></a>, <a class="ref" href="instalg1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E8:52_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="instalg1.html#E6:52_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E5:52_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:52_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="instalg1.html#S1" target="_self">INSTALG1:sch 1</a>(<a class="txt" href="instalg1.html#E3:52_1_1"><i><font color="Green">E29</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E8:52_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="instalg1.html#E5:52_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="instalg1.html#E6:52_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div>
