<?xml version="1.0"?>
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<b>let </b><font color="Maroon" title="c1">S</font> be  <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:()"><i><font color="Red">::  thesis: </font></i></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="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">R1</font>, <font color="Olive" title="b3">R2</font> being  <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b4">t</font> being   <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="Olive" title="b1">X</font></span>)</span>  st <font color="Olive" title="b2">R1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">R2</font> holds <br/> <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Olive" title="b2">R1</font>,<font color="Olive" title="b4">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Olive" title="b3">R2</font>,<font color="Olive" title="b4">t</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being  <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</a> of  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b3">t</font> being   <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>  st <font color="Olive" title="b1">R1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">R2</font> holds <br/> <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Olive" title="b1">R1</font>,<font color="Olive" title="b3">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Olive" title="b2">R2</font>,<font color="Olive" title="b3">t</font></span><br/><b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font> be   <a href="osalg_4.html#NM4" title="OSALG_4:NM.4">OSCongruence</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <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>  st <font color="Maroon" title="c3">R1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">R2</font> holds <br/> <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Olive" title="b1">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Olive" title="b1">t</font></span><br/><b>let </b><font color="Maroon" title="c5">t</font> be    <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">R1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">R2</font> implies  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font> )</span><br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">R2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font></span><br/>

<b>set </b><font color="Maroon" title="c6">s</font> =  <a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font>;<br/>
<b>set </b><font color="Maroon" title="c7">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c8">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/>
<b>set </b><font color="Maroon" title="c9">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">t</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font> )</span><br/>

<b>assume </b><a NAME="E2:76"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">xa</font> = <font color="Maroon" title="c5">t</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> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#D12" target="_self" title="OSAFREE:def.12">Def12</a></i>;<br/>
<i><font color="Green" title="E54">A3</font></i>:  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c5">t</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c3">R1</font>,<font color="Maroon" title="c11">xa</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p3">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">xa</font>

;<br/>
<b>set </b><font color="Maroon" title="c12">CC1</font> =  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">CR1</font> =  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:76"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">xa</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">R1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p2">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:76"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E4:76"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E7:76"/><i><font color="Green" title="E55">A4</font></i>: 
( <font color="Maroon" title="c14">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <span class="p1">(<span class="default"><a href="osafree.html#K10" title="OSAFREE:func.10">LeastSort</a> <font color="Maroon" title="c5">t</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">xa</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="c3">R1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c14">s1</font> )
 <b>by </b><i><a class="ref" href="osalg_4.html#D11" title="OSALG_4:def.11">OSALG_4:def 11</a></i>;<br/>
<a NAME="E8:76"/><i><font color="Green" title="E56">A5</font></i>: 
( <font color="Maroon" title="c10">x</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="c14">s1</font> &amp; <font color="Maroon" title="c11">xa</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="c14">s1</font> )
 
<b>by </b><i><a class="txt" href="#E7:76"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">xa</font> = <font color="Maroon" title="c5">t</font>, <font color="Maroon" title="c16">xb</font> = <font color="Maroon" title="c10">x</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="c14">s1</font> <b>by </b><i><a class="txt" href="#E7:76"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">t1</font> = <font color="Maroon" title="c10">x</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> <b>by </b><i><a class="txt" href="#E8:76"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="osafree.html#T16" target="_self" title="OSAFREE:th.16">Th16</a></i>;<br/>
<a NAME="E11:76"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c14">s1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R2</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c14">s1</font>
 
<b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c17">t1</font> = 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c16">xb</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
<br/>.= 
 <a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c15">xa</font>
<b>by </b><i><a class="txt" href="#E7:76"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="#E11:76"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="osalg_4.html#T13" title="OSALG_4:th.13">OSALG_4:13</a></i>
<br/>.= 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font>
<b>by </b><i><a class="ref" href="osafree.html#D28" target="_self" title="OSAFREE:def.28">Def28</a></i>
;<br/>
<b>hence </b><a NAME="E13:76"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <font color="Maroon" title="c4">R2</font>,<font color="Maroon" title="c5">t</font>
 <b>by </b><i><a class="ref" href="osafree.html#T35" target="_self" title="OSAFREE:th.35">Th35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
