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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a>   <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a>   <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a>  <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">V16</a>() <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">R</font> being   <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>   <a href="osalg_4.html#V3" title="OSALG_4:attr.3">monotone</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Olive" title="b1">X</font> holds   <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V16</a>() <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>   <a href="osalg_4.html#V3" title="OSALG_4:attr.3">monotone</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font> holds   <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">R</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">SPTA</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">M</font> =  <a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">P</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">O1</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">R</font> be   <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>   <a href="osalg_4.html#V3" title="OSALG_4:attr.3">monotone</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c9">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">i</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="c1">S</font> or <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:90"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c10">i</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="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90"><span class="lab"><font color="Green" title="E63">A1</font></span></a></span>;<br/>
<a NAME="E4:90"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c9">R</font> is  <a href="osalg_4.html#V1" title="OSALG_4:attr.1">os-compatible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_4.html#D2" title="OSALG_4:def.2">OSALG_4:def 2</a></span>;<br/>
<a NAME="E5:90"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:90_1"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></span>;<br/>
<a NAME="E2:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A4</font></span>: 
<font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>;<br/>
<a NAME="E3:90_1"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</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">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:90_1"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</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">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">ta</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">tb</font> = <font color="Maroon" title="c13">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E6:90_1"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E7:90_1"/><span class="lab"><font color="Green" title="E70">A8</font></span>: 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c14">ta</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c15">tb</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="osalg_4.html#T12" title="OSALG_4:th.12">OSALG_4:12</a></span>;<br/>
<a NAME="E8:90_1"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E9:90_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">t1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c17">t2</font> = <font color="Maroon" title="c13">b</font> as   <a href="osafree.html#NM2" title="OSAFREE:NM.2">Element</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">t1</font> = <font color="Maroon" title="c16">t1</font>, <font color="Maroon" title="c19">t2</font> = <font color="Maroon" title="c17">t2</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T14" target="_self" title="OSAFREE:th.14">Th15</a></span>;<br/>
<a NAME="E12:90_1"/><span class="lab"><font color="Green" title="E72">A10</font></span>: 
<font color="Maroon" title="c19">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c19">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E13:90_1"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E14:90_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c19">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T40" target="_self" title="OSAFREE:th.40">Th41</a></span>;<br/>
<a NAME="E15:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A12</font></span>: 
<font color="Maroon" title="c8">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c19">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D16" title="OSALG_1:def.16">OSALG_1:def 16</a></span>;<br/>
<a NAME="E16:90_1"/><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">t2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</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> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c19">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T44" target="_self" title="OSAFREE:th.44">Th45</a></span>;<br/>
<a NAME="E17:90_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c19">t2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E18:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">t2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</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> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E12:90_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E13:90_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E15:90_1"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E16:90_1"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="ref" href="osalg_4.html#D1" title="OSALG_4:def.1">OSALG_4:def 1</a></span>;<br/>
<a NAME="E19:90_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E20:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font></span>)</span>,<font color="Maroon" title="c19">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="txt" href="#E18:90_1"><span class="lab"><font color="Green" title="E76">A14</font></span></a>, <a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></span>;<br/>
<a NAME="E21:90_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c18">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T40" target="_self" title="OSAFREE:th.40">Th41</a></span>;<br/>
<a NAME="E22:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A16</font></span>: 
<font color="Maroon" title="c8">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c18">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D16" title="OSALG_1:def.16">OSALG_1:def 16</a></span>;<br/>
<a NAME="E23:90_1"/><span class="lab"><font color="Green" title="E79">A17</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E24:90_1"/><span class="lab"><font color="Green" title="E80">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">t1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</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> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c18">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T44" target="_self" title="OSAFREE:th.44">Th45</a></span>;<br/>
<a NAME="E25:90_1"/><span class="lab"><font color="Green" title="E81">A19</font></span>: 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c19">t2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c15">tb</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D27" target="_self" title="OSAFREE:def.27">Def28</a></span>;<br/>
<a NAME="E26:90_1"/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c18">t1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c14">ta</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D27" target="_self" title="OSAFREE:def.27">Def28</a></span>;<br/>
<a NAME="E27:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> (<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c19">t2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E25:90_1"><span class="lab"><font color="Green" title="E81">A19</font></span></a>, <a class="ref" href="osafree.html#T34" target="_self" title="OSAFREE:th.34">Th35</a></span>;<br/>
<a NAME="E28:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A20</font></span>: 
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T42" target="_self" title="OSAFREE:th.42">Th43</a></span>;<br/>
<a NAME="E29:90_1"/><span class="lab"><font color="Green" title="E83">A21</font></span>: 
<font color="Maroon" title="c18">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c18">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E30:90_1"/>
 <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c18">t1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></span>;<br/>
<a NAME="E31:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">t1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</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> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E29:90_1"><span class="lab"><font color="Green" title="E83">A21</font></span></a>, <a class="txt" href="#E23:90_1"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="txt" href="#E22:90_1"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="txt" href="#E24:90_1"><span class="lab"><font color="Green" title="E80">A18</font></span></a>, <a class="ref" href="osalg_4.html#D1" title="OSALG_4:def.1">OSALG_4:def 1</a></span>;<br/>
<a NAME="E32:90_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:90_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">R</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="txt" href="#E28:90_1"><span class="lab"><font color="Green" title="E82">A20</font></span></a>, <a class="txt" href="#E31:90_1"><span class="lab"><font color="Green" title="E84">A22</font></span></a>, <a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E20:90_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:90"/>
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
