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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="b3">F</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b4">C</font>, <font color="Olive" title="b5">E</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b6">D</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b4">C</font>  st <font color="Olive" title="b5">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b6">D</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b4">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b6">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b5">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">E</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b4">D</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b2">C</font>  st <font color="Olive" title="b3">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">D</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b4">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b3">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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">C</font>, <font color="Olive" title="b2">E</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">D</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b2">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b3">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b2">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">E</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">A</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">D</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c5">C</font>  st <font color="Maroon" title="c6">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b1">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">D</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c5">C</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="c6">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">D</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c6">E</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">D</font>
 ; <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"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><a NAME="E2:6_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><a NAME="E3:6_1"/>
<font color="Maroon" title="c7">D</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/><a NAME="E4:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></span>;<br/><a NAME="E5:6_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">fc</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E5:6_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c9">fc</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="msafree.html#D1" title="MSAFREE:def.1">MSAFREE:def 1</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">F</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">E</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
