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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">I0</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font><br/> for <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">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font><br/> for <font color="Olive" title="b5">A0</font>, <font color="Olive" title="b6">B0</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">I0</font>  st <font color="Olive" title="b5">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Olive" title="b1">I0</font> &amp; <font color="Olive" title="b6">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Olive" title="b1">I0</font> holds <br/><font color="Olive" title="b4">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Olive" title="b1">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b5">A0</font>,<font color="Olive" title="b6">B0</font></span><br/><b>let </b><font color="Maroon" title="c2">I0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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">A</font>, <font color="Olive" title="b2">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="b3">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b4">A0</font>, <font color="Olive" title="b5">B0</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I0</font>  st <font color="Olive" title="b4">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> &amp; <font color="Olive" title="b5">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> holds <br/><font color="Olive" title="b3">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b4">A0</font>,<font color="Olive" title="b5">B0</font></span><br/><b>let </b><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">F</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font><br/> for <font color="Olive" title="b2">A0</font>, <font color="Olive" title="b3">B0</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I0</font>  st <font color="Olive" title="b2">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> &amp; <font color="Olive" title="b3">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> holds <br/><font color="Olive" title="b1">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b2">A0</font>,<font color="Olive" title="b3">B0</font></span><br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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">A0</font>, <font color="Olive" title="b2">B0</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I0</font>  st <font color="Olive" title="b1">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> &amp; <font color="Olive" title="b2">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Olive" title="b1">A0</font>,<font color="Olive" title="b2">B0</font></span><br/><b>let </b><font color="Maroon" title="c6">A0</font>, <font color="Maroon" title="c7">B0</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> &amp; <font color="Maroon" title="c7">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> implies <font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">A0</font>,<font color="Maroon" title="c7">B0</font> )</span><br/>









<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c6">A0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> &amp; <font color="Maroon" title="c7">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">B</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">A0</font>,<font color="Maroon" title="c7">B0</font></span><br/>

<a NAME="E2:9"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green" title="E8">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:9"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c2">I0</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I0</font> implies <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c7">B0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/><b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c7">B0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><a NAME="E2:9_1"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="#E4:9"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E3:9_1"/>
( <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E3:9"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="#E1:9_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c7">B0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="#E1:9_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="#E2:9_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</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="E7:9"/>
<font color="Maroon" title="c5">F</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <font color="Maroon" title="c2">I0</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">A0</font>,<font color="Maroon" title="c7">B0</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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