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<span class="kw">let </span><font color="Maroon" title="c1">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font> holds <br/><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">B</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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"> ( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font> implies <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">B</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</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="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">assume </span><a NAME="E4:17"/>
<a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">B</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:17"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:17"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">e</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">d</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:17"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E11:17"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E12:17"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E10:17"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="jordan6.html#T31" onmouseover="rs('jordan6/T31')" onmouseout="rh()">JORDAN6:31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:17"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E15:17"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<a NAME="E16:17"/>
<font color="Maroon" title="c7">c</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E14:17"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="jordan6.html#T31" onmouseover="rs('jordan6/T31')" onmouseout="rh()">JORDAN6:31</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c7">c</font> = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<font color="Maroon" title="c5">e</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="pscomp_1.html#D6" onmouseover="rs('pscomp_1/D6')" onmouseout="rh()">PSCOMP_1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">d</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="pscomp_1.html#D6" onmouseover="rs('pscomp_1/D6')" onmouseout="rh()">PSCOMP_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:17"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E14:17"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E10:17"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</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/>


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