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<b>let </b><font color="Maroon">G1</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,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">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>, <font color="Maroon">G2</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>,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">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:52_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">G1</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">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:52_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds  <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</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">x</font>,<font color="Olive">s</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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E3:52_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">G2</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">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:52_1_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds  <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</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">x</font>,<font color="Olive">s</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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">H1</font> = <font color="Maroon">G1</font>;<br/>
<b>set </b><font color="Maroon">H2</font> = <font color="Maroon">G2</font>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S1</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S1</font> ;<br/><b>reconsider </b><font color="Maroon">g1</font> = <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">g2</font> = <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><a NAME="E1:52_1_2_1_1"/>
 <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></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/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_2_1_1"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">z</font> <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="Maroon">a</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G1</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">z</font>
<b>by </b><i/>
<br/>.= 
 <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</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G2</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">z</font>
<b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:52_1_2"/><b>then </b>
( <font color="Maroon">G1</font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">G2</font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f</font>,<font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E8:52_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , <a class="ref" href="extens_1.html#T18">EXTENS_1:18</a></i>;<br/>


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