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<b>let </b><font color="Maroon" title="c3">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">C</font> being <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Olive" title="b1">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q</font> iff  ex <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b5">p'</font>, <font color="Olive" title="b6">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b4">i</font></span>)</span> st <br/>( <font color="Olive" title="b5">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b6">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b5">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b6">q'</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">C</font> be  <a href="pcs_0.html#V16" title="PCS_0:attr.16">V208</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">q</font> iff  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b4">p'</font>, <font color="Olive" title="b5">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b3">i</font></span>)</span> st <br/>( <font color="Olive" title="b4">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b5">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b4">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b5">q'</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K17" title="PCS_0:func.17">pcs-union</a> <font color="Maroon" title="c4">C</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="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> iff  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> ) )</span><br/>





<b>thus </b><a NAME="E1:116"/>
( <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> ) implies <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_1"/>
<font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> )</span><br/>

<a NAME="E2:116_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">P</font> being   <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a>  ex <font color="Olive" title="b3">p'</font>, <font color="Olive" title="b4">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">P</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b4">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b3">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b4">q'</font> )
 
<b>by </b><i><a class="ref" href="pcs_0.html#T12" target="_self" title="PCS_0:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:116_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:116"/>
(  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">I</font> ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="pcs_0.html#K15" title="PCS_0:func.15">.</a> <font color="Olive" title="b1">i</font></span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Olive" title="b2">p'</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b3">q'</font> ) implies <font color="Maroon" title="c5">p</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">q</font> )
 <b>by </b><i><a class="ref" href="pcs_0.html#T12" target="_self" title="PCS_0:th.12">Th12</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>
