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

<b>let </b><font color="Maroon">S1</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> , <font color="Maroon">S2</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> , <font color="Maroon">S3</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">X</font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S3</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f1</font>,<font color="Maroon">g1</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">f2</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S3</font>;<br/><b>let </b><font color="Maroon">g2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>set </b><font color="Maroon">f</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>;<br/>
<b>assume </b><a NAME="E2:57"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f2</font>,<font color="Maroon">g2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S2</font>,<font color="Maroon">S3</font>
 ;<br/>

<a NAME="E3:57"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font>,<font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">S1</font>,<font color="Maroon">S3</font>
 
<b>by </b><i>, <a class="ref" href="pua2mss1.html#T30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <a href="instalg1.html#K3" target="_self">hom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S3</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> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></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>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> &amp;  <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f1</font></span>)</span>,<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">f2</font></span>)</span></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f1</font>,<font color="Maroon">g1</font> &amp;  <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font>,<font color="Maroon">X</font>,<font color="Maroon">S2</font>,<font color="Maroon">S3</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">f2</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">S2</font>,<font color="Maroon">f2</font>,<font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:57"/><i><font color="Green">E56</font></i>: 
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">f2</font></span>)</span></span>)</span> <a href="pzfmisc1.html#R1">is_transformable_to</a> 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">S2</font>,<font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S2</font> ;<br/>
<a NAME="E3:57_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">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S2</font>,<font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E4:57_1"/>
(  not 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">S2</font>,<font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or 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">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">Af</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>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>, <font color="Maroon">Af1</font> = <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">f2</font></span>)</span></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f1</font>,<font color="Maroon">g1</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/>
<a NAME="E7:57"/>
<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>,<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <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">S2</font>,<font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<font color="Maroon">f1</font>,<font color="Maroon">g1</font>
 
<b>by </b><i>, <a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">hf1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">Af1</font>,<font color="Maroon">Af</font><b> such that </b><br/><a NAME="E9:57"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">hf1</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font>,<font color="Maroon">X</font>,<font color="Maroon">S2</font>,<font color="Maroon">S3</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">hf1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">Af1</font>,<font color="Maroon">Af</font> )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E10:57"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">hh</font> =  <a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">S1</font>,<font color="Maroon">S2</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">X</font> <a href="pboole.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Af1</font> <b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">hf1h</font> = <font color="Maroon">hf1</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">hh</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">X</font> <a href="pboole.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></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">X</font></span>)</span> <a href="instalg1.html#K1" target="_self">|</a> <font color="Maroon">S1</font>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span></span>)</span> ;<br/>
<a NAME="E13:57"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">hf1</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">hh</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> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<font color="Maroon">Af</font>
 
<b>by </b><i>, , <a class="txt" href="instalg1.html#E1:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </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">S1</font>;<br/><b>let </b><font color="Maroon">x</font> be    <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> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><a NAME="E1:57_2"/><i><font color="Green">E105</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="instalg1.html#E1:57"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>reconsider </b><font color="Maroon">t</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">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> as   <a href="msaterm.html#NM1">Term</a> of <font color="Maroon">S1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#T4">MSATERM:4</a></i>;<br/><a NAME="E3:57_2"/>
(  <a href="msaterm.html#K7">the_sort_of</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K7">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<font color="Maroon">s</font> &amp;  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> #) )
 <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>, <a class="ref" href="msaterm.html#T14">MSATERM:14</a></i>;<br/><a NAME="E4:57_2"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> 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> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="msaterm.html#D5">MSATERM:def 5</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">hf1h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">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="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">hf1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></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="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">hf1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E1:57"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">hf1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">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="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="instalg1.html#E1:57"><i><font color="Green">E37</font></i></a>, </i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">hf1h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">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="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font>,<font color="Maroon">X</font>,<font color="Maroon">S2</font>,<font color="Maroon">S3</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></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="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></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">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="instalg1.html#T2" target="_self">Th2</a>, , </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">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <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><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:57"/>
 <a href="instalg1.html#K3" target="_self">hom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>,<font color="Maroon">X</font>,<font color="Maroon">S1</font>,<font color="Maroon">S3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font>,<font color="Maroon">X</font>,<font color="Maroon">S2</font>,<font color="Maroon">S3</font></span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">f1</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="instalg1.html#K3" target="_self">hom</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


</div>
