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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b2">SF</font> being   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font><br/>  for <font color="Olive" title="b3">V</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b3">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> holds <br/> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Olive" title="b2">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><br/>  for <font color="Olive" title="b2">V</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b2">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> holds <br/> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Olive" title="b1">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>  st <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b1">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> holds <br/> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">V</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> implies  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">V</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:75"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">V</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">V</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:75_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c5">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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="c5">i</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:75_1"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E4:75_1"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c6">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#D1" target="_self" title="MSSUBFAM:def.1">Def2</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E4:75_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="setfam_1.html#D9" title="SETFAM_1:def.9">SETFAM_1:def 9</a></span>
<br/>.= 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E1:75_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:75_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:75"/>
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">SF</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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