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

<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="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="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">t1</font>, <font color="Olive" title="b3">t2</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> holds <br/> ( <font color="Olive" title="b3">t2</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="Olive" title="b1">X</font></span>)</span>,<font color="Olive" title="b2">t1</font> iff <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">t2</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="Olive" title="b1">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t1</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">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</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> holds <br/> ( <font color="Olive" title="b2">t2</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="Olive" title="b1">t1</font> iff <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="Olive" title="b2">t2</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="Olive" title="b1">t1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">t1</font>, <font color="Maroon" title="c4">t2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">t2</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="c3">t1</font> iff <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="c4">t2</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="c3">t1</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</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">thus </span><a NAME="E1:88"/>
( <font color="Maroon" title="c4">t2</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="c3">t1</font> implies <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="c4">t2</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="c3">t1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T42" target="_self" title="OSAFREE:th.42">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c4">t2</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="c3">t1</font> implies <font color="Maroon" title="c4">t2</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="c3">t1</font> )</span><br/>

<a NAME="E2:88"/>
<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="c4">t2</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="c4">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T41" target="_self" title="OSAFREE:th.41">Th41</a></span>;<br/>
<a NAME="E3:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A1</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="c4">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<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="c4">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T35" target="_self" title="OSAFREE:th.35">Th35</a></span>;<br/>
<a NAME="E4:88"/>
<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="c3">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="c3">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T41" target="_self" title="OSAFREE:th.41">Th41</a></span>;<br/>
<a NAME="E5:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A2</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="c3">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<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="c3">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osafree.html#T35" target="_self" title="OSAFREE:th.35">Th35</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:88"/>
<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="c4">t2</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="c3">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="c4">t2</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="c3">t1</font></span><br/>

<span class="kw">hence </span><a NAME="E7:88"/>
<font color="Maroon" title="c4">t2</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="c3">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="osafree.html#T33" target="_self" title="OSAFREE:th.33">Th33</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>
