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<span class="kw">let </span><font color="Maroon" title="c1">A</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</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>  st  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> holds <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font> &amp; <font color="Olive" title="b2">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">q1</font>, <font color="Maroon" title="c3">q2</font>, <font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> implies ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font> ) )</span><br/>









<a NAME="E1:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c3">q2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D5" title="JORDAN6:def.5">JORDAN6:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font> &amp; <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font> )</span><br/>

<a NAME="E3:10"/>
 <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</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> :  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c3">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D3" title="JORDAN6:def.3">JORDAN6:def 3</a></span>;<br/>
<a NAME="E4:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c3">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E5:10"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<a NAME="E6:10"/><span class="kw">then </span>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T8" target="_self" title="JORDAN16:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:10"/>
<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font></span><br/>

<a NAME="E8:10"/>
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</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> :  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">q1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D4" title="JORDAN6:def.4">JORDAN6:def 4</a></span>;<br/>
<a NAME="E9:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E10:10"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<a NAME="E11:10"/><span class="kw">then </span>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c3">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T7" target="_self" title="JORDAN16:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:10"/>
<font color="Maroon" title="c3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">q1</font>,<font color="Maroon" title="c3">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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