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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <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>; <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">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Olive" title="b1">s</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</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="Olive" title="b1">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font> implies  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</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="c3">s</font> )</span><br/>





<a NAME="E1:15"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <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> : <font color="Olive" title="b1">p1</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="c3">s</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D6" onmouseover="rs('jordan6/D6')" onmouseout="rh()">JORDAN6:def 6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:15"/>
( <font color="Maroon" title="c1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</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="c3">s</font></span><br/>

<a NAME="E3:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( <font color="Maroon" title="c1">p</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="c3">s</font> &amp; <font color="Maroon" title="c2">q</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="c3">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T31" onmouseover="rs('jordan6/T31')" onmouseout="rh()">JORDAN6:31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>) or <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:15"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</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"> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c4">u</font> as   <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> ;<br/>
<a NAME="E6:15"/>
<font color="Maroon" title="c5">p1</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="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="goboard7.html#T5" onmouseover="rs('goboard7/T5')" onmouseout="rh()">GOBOARD7:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:15"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E8">A1</font></span></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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