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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E1</font>, <font color="Olive" title="b2">E2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/>  for <font color="Olive" title="b3">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">E2</font> holds <br/>(  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b1">E1</font>,<font color="Olive" title="b3">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b2">E2</font>,<font color="Olive" title="b3">p</font>) &amp;  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b1">E1</font>,<font color="Olive" title="b3">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b2">E2</font>,<font color="Olive" title="b3">p</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">E1</font>, <font color="Maroon" title="c4">E2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c3">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">E2</font> holds <br/>(  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Olive" title="b1">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Olive" title="b1">p</font>) &amp;  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Olive" title="b1">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Olive" title="b1">p</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">E2</font> implies (  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) &amp;  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c3">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">E2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) &amp;  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds  not <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">q</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span></span><br/><span class="kw">assume </span><a NAME="E1:43_1"/>
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
( <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) &amp; <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 ex <font color="Olive" title="b1">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E1</font> )
 ;<br/><a NAME="E4:43_1"/>
 ex <font color="Olive" title="b1">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E3:43"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:43"/>
 <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds  not <font color="Olive" title="b1">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">q</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span></span><br/><span class="kw">assume </span><a NAME="E1:43_2"/>
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
( <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) &amp; <font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 ex <font color="Olive" title="b1">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E1</font> )
 ;<br/><a NAME="E4:43_2"/>
 ex <font color="Olive" title="b1">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x2</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E6:43"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)</span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E7:43"/>
 <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c3">E1</font>,<font color="Maroon" title="c5">p</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c4">E2</font>,<font color="Maroon" title="c5">p</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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