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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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</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="Olive" title="b1">X</font></span>)</span><br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b3">t</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="Olive" title="b4">x</font>,<font color="Olive" title="b2">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b5">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Olive" title="b1">X</font></span>)</span>,<font color="Olive" title="b3">t</font> iff <font color="Olive" title="b5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">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><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b2">t</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="Olive" title="b3">x</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b4">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Olive" title="b2">t</font> iff <font color="Olive" title="b4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> )</span><br/><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">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><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Olive" title="b1">t</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="Olive" title="b2">x</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b3">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Olive" title="b1">t</font> iff <font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> )</span><br/><b>let </b><font color="Maroon" title="c4">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"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c4">t</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="Olive" title="b1">x</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Olive" title="b2">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> iff <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )</span><br/>







<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">R1</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E1:77"/>
 <a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D18" target="_self" title="OSAFREE:def.18">Def18</a></i>;<br/>
<a NAME="E2:77"/><b>then </b><i><font color="Green" title="E53">A1</font></i>: 
 <a href="osafree.html#K25" title="OSAFREE:func.25">OSClass</a> <span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">t</font>
 
<b>by </b><i><a class="ref" href="osafree.html#T36" target="_self" title="OSAFREE:th.36">Th36</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c4">t</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="c7">x</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies ( <font color="Maroon" title="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> iff <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> ) )</span><br/>

<b>assume </b><a NAME="E3:77"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c4">t</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="c7">x</font>,<font color="Maroon" title="c3">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> iff <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )</span><br/>

<b>thus </b><a NAME="E4:77"/>
( <font color="Maroon" title="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> implies <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E2:77"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="osafree.html#E3:77"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="osafree.html#T34" target="_self" title="OSAFREE:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> implies <font color="Maroon" title="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font> )</span><br/>

<b>assume </b><a NAME="E5:77"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font></span><br/>

<b>hence </b><a NAME="E6:77"/>
<font color="Maroon" title="c8">x1</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#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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