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<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</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">X</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <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="c2">S2</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> holds <br/> <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <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="c2">S2</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> holds <br/> <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</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> being  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> holds <br/> <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> implies  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c6">A</font> = <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">H</font> =  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">t1</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> st <br/>( <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b2">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t2</font> holds <br/><font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t2</font> ) );<br/>
<a NAME="E1:55"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeOper</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:55"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c5">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="instalg1.html#T10" target="_self" title="INSTALG1:th.10">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">B</font> = <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> as   <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">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="instalg1.html#T23" target="_self" title="INSTALG1:th.23">Th23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">H9</font> =  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">B</font> ;<br/>
<a NAME="E6:55"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st ( for <font color="Olive" title="b3">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</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">p</font> being   <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st ( for <font color="Olive" title="b2">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b1">p</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">p</font> be    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>; <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">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">p</font>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:55_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</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">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">p</font>]</span><br/>

<a NAME="E2:55_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c12">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></span>;<br/>
<a NAME="E4:55_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">q</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c13">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="instalg1.html#T25" target="_self" title="INSTALG1:th.25">Th25</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c15">t1</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c11">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c12">p</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="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">p</font> &amp; ( for <font color="Olive" title="b1">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t2</font> holds <br/><font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:55_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></span>; <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">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t2</font> holds <br/><font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">q</font> as    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">t2</font> be   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t2</font> implies <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:55_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
(  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t2</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="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font></span><br/>

<a NAME="E9:55_1"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a></span>;<br/>
<a NAME="E10:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c14">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E6:55_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="instalg1.html#T41" target="_self" title="INSTALG1:th.41">Th41</a></span>;<br/>
<a NAME="E11:55_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c16">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></span>;<br/>
<a NAME="E12:55_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c16">b</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></span>;<br/>
<a NAME="E13:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">t2</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E8:55_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="instalg1.html#T42" target="_self" title="INSTALG1:th.42">Th42</a></span>;<br/>
<a NAME="E14:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">o9</font>, <font color="Maroon" title="c19">s9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:55_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c18">o9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>
 <span class="kw">and </span><br/><a NAME="E17:55_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
( <font color="Maroon" title="c19">s9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c17">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">o9</font>,<font color="Maroon" title="c19">s9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">o9</font> = <font color="Maroon" title="c18">o9</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:55_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a></span>;<br/>
<a NAME="E19:55_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c17">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o9</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:55_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">r</font> being    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c20">o9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E21:55_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c17">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o9</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T10" title="MSATERM:th.10">MSATERM:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">c</font> = <font color="Maroon" title="c21">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o9</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="instalg1.html#T2" target="_self" title="INSTALG1:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">R</font> = <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c22">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">o9</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="instalg1.html#T25" target="_self" title="INSTALG1:th.25">Th25</a></span>;<br/>
<a NAME="E24:55_1"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c20">o9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:55_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="msaterm.html#T17" title="MSATERM:th.17">MSATERM:17</a></span>;<br/>
<a NAME="E25:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o9</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c23">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E8:55_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E21:55_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="instalg1.html#T41" target="_self" title="INSTALG1:th.41">Th41</a></span>;<br/>
<a NAME="E26:55_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o9</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></span>;<br/>
<a NAME="E27:55_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">o9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E28:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">o9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<a NAME="E29:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></span>;<br/>
<a NAME="E30:55_1"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E25:55_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span> implies <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> )</span><br/><a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
<font color="Maroon" title="c23">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H9</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c22">c</font>
 ;<br/><span class="kw">assume </span><a NAME="E2:55_1_1"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</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="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">w1</font> = <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>, <font color="Maroon" title="c26">w2</font> = <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E29:55_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></span>;<br/><a NAME="E4:55_1_1"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="ref" href="msaterm.html#T23" title="MSATERM:th.23">MSATERM:23</a></span>;<br/><a NAME="E5:55_1_1"/>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H9</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c13">a</font>
 ;<br/><a NAME="E6:55_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E4:55_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></span>;<br/><a NAME="E7:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c25">w1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:55_1"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E28:55_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E29:55_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E4:55_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></span>;<br/><a NAME="E8:55_1_1"/>
<font color="Maroon" title="c25">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:55_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A22</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c25">w1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>;<br/><a NAME="E10:55_1_1"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c26">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:55_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E29:55_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E2:55_1_1"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="ref" href="msaterm.html#T23" title="MSATERM:th.23">MSATERM:23</a></span>;<br/><span class="kw">hence </span><a NAME="E11:55_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> <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="txt" href="#E9:55_1_1"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E4:55_1_1"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E7:55_1_1"><span class="lab"><font color="Green" title="E51">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="E32:55_1"/>
<font color="Maroon" title="c15">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E21:55_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E28:55_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E29:55_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</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">thus </span><a NAME="E7:55"/>
 <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="instalg1.html#K1" title="INSTALG1:func.1">|</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">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="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</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">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c11">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> <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> or <font color="Maroon" title="c12">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E8:55"/>
<font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> or <font color="Maroon" title="c12">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c11">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</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/>
<span class="kw">assume </span><a NAME="E10:55"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
<span class="p1">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</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="c12">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A24</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">H9</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c13">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</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="c14">x</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="c12">h</font> or  not <font color="Maroon" title="c15">y</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="c12">h</font> or  not <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:55"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
( <font color="Maroon" title="c14">x</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="c12">h</font> &amp; <font color="Maroon" title="c15">y</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="c12">h</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="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )</span><br/>

<a NAME="E13:55"/>
 <a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c13">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">S1</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T12" title="MSATERM:th.12">MSATERM:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">t1</font> = <font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c17">t2</font> = <font color="Maroon" title="c15">y</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:55"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E36">A25</font></span></a></span>;<br/>
<a NAME="E15:55"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</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">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c19">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c20">t1</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T4" title="MSATERM:th.4">MSATERM:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c20">t1</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="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( for <font color="Olive" title="b1">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t2</font> holds <br/><font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:55_3"/>
<font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <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">t2</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>  st  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t2</font> holds <br/><font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c21">t2</font> be   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c21">t2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t2</font> implies <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">t2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:55_3"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c21">t2</font>
 <span class="kw">and </span><br/><a NAME="E4:55_3"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t2</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="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">t2</font></span><br/>

<a NAME="E5:55_3"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T14" title="MSATERM:th.14">MSATERM:14</a></span>;<br/>
<a NAME="E6:55_3"/><span class="lab"><font color="Green" title="E41">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<font color="Maroon" title="c18">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:55_3_1"/>
<font color="Maroon" title="c21">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S1</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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="E2:55_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">t2</font> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></span>;<br/><a NAME="E3:55_3_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="msaterm.html#M2" title="MSATERM:mode.2">CompoundTerm</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E3:55_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E4:55_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E6:55_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E5:55_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="instalg1.html#T42" target="_self" title="INSTALG1:th.42">Th42</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/><a NAME="E4:55_3_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">S2</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#D6" title="MSATERM:def.6">MSATERM:def 6</a></span>;<br/><a NAME="E5:55_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A31</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></span>;<br/><a NAME="E6:55_3_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</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="c2">S2</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:55_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55_3_1"><span class="lab"><font color="Green" title="E42">A31</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">then </span><span class="kw">consider </span><font color="Maroon" title="c22">s2</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c23">v2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s2</font><span class="kw"> such that </span><br/><a NAME="E9:55_3"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c21">t2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">v2</font>,<font color="Maroon" title="c22">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T2" title="MSATERM:th.2">MSATERM:2</a></span>;<br/>
<a NAME="E10:55_3"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c21">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">v2</font>,<font color="Maroon" title="c22">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55_3"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="msaterm.html#T5" title="MSATERM:th.5">MSATERM:5</a></span>;<br/>
<a NAME="E11:55_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A34</font></span>: 
<font color="Maroon" title="c18">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E5:55_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="msaterm.html#T14" title="MSATERM:th.14">MSATERM:14</a></span>;<br/>
<a NAME="E12:55_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="instalg1.html#K3" title="INSTALG1:func.3">hom</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E10:55_3"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="instalg1.html#D5" target="_self" title="INSTALG1:def.5">Def5</a></span>;<br/>
<a NAME="E13:55_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">v</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E6:55_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E5:55_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="ref" href="trees_4.html#T4" title="TREES_4:th.4">TREES_4:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:55_3"/>
<font color="Maroon" title="c20">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">t2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55_3"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E11:55_3"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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>
<a NAME="E16:55"/>
for <font color="Olive" title="b1">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="msaterm.html#S1" title="MSATERM:sch.1">MSATERM:sch 1</a>(<a class="txt" href="#E15:55"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E33">A3</font></span></a>);<br/></span>
<a NAME="E17:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A35</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">t1</font>]
 
;<br/>
<a NAME="E18:55"/>
(  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:55"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="txt" href="#E12:55"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="msaterm.html#D5" title="MSATERM:def.5">MSATERM:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:55"/>
(  not <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E17:55"><span class="lab"><font color="Green" title="E38">A35</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>
