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<b>let </b><font color="Maroon" title="c1">I</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"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b4">SF</font> being  <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b4">SF</font> &amp; <font color="Olive" title="b2">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b3">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b4">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b3">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span><br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">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">SF</font> being  <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">SF</font> &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b1">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span><br/><b>let </b><font color="Maroon" title="c5">SF</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c5">SF</font> &amp; <font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c5">SF</font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><b>then consider </b><font color="Maroon" title="c7">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span><b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c7">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>and </b><br/><a NAME="E4:71_1"/><i><font color="Green" title="E18">A5</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" target="_self" title="MSSUBFAM:def.2">Def2</a></i>;<br/><a NAME="E5:71_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="#E1:71"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E1:71_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/><a NAME="E6:71_1"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="txt" href="#E3:71_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><a NAME="E7:71_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="#E2:71"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E1:71_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="pboole.html#D8" title="PBOOLE:def.8">PBOOLE:def 8</a></i>;<br/><a NAME="E8:71_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="pboole.html#T5" title="PBOOLE:th.5">PBOOLE:5</a></i>;<br/><a NAME="E9:71_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E10:71_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="#E3:71_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E5:71_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="setfam_1.html#T9" title="SETFAM_1:th.9">SETFAM_1:9</a></i>;<br/><a NAME="E11:71_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:71_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="ref" href="pboole.html#T5" title="PBOOLE:th.5">PBOOLE:5</a></i>;<br/><b>hence </b><a NAME="E13:71_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="#E1:71_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E4:71_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E6:71_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="pboole.html#D8" title="PBOOLE:def.8">PBOOLE:def 8</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><b>end;</b></div>
<b>hence </b><a NAME="E4:71"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c5">SF</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
