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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X1</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y1</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Olive" title="b1">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Olive" title="b1">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Olive" title="b1">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y1</font> )</span><br/><b>let </b><font color="Maroon" title="c4">Y1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y1</font> ) )</span><br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y1</font> )</span><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#T9" target="_self" title="BORSUK_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c3">X1</font>
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T12" target="_self" title="BORSUK_1:th.12">Th12</a></i>
; <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="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y1</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#T10" target="_self" title="BORSUK_1:th.10">Th10</a></i>
<br/>.= 
<font color="Maroon" title="c4">Y1</font>
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T12" target="_self" title="BORSUK_1: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/>


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