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<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">d</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">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><br/>  for <font color="Olive" title="b3">P</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="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp;  not <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp; <font color="Olive" title="b3">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> holds <br/> <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c11">p1</font>, <font color="Maroon" title="c12">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</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="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp;  not <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp; <font color="Olive" title="b1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> holds <br/> <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c13">P</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp;  not <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) &amp; <font color="Maroon" title="c13">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> implies  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) )</span><br/>













<span class="kw">set </span><font color="Maroon" title="c14">R</font> =  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c15">dR</font> =  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c16">n</font> =  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c9">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">d</font>
 <span class="kw">and </span><br/><a NAME="E3:82"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">and </span><br/><a NAME="E4:82"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
 not <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">and </span><br/><a NAME="E5:82"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</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="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:82"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c17">x</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="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E7:82"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
 not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A6</font></span></a></span>;<br/>
<a NAME="E9:82"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="jordan.html#T52" target="_self" title="JORDAN:th.52">Th52</a></span>;<br/>
<a NAME="E10:82"/>
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<a NAME="E11:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E12:82"/>
<font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<a NAME="E13:82"/><span class="kw">then </span>
<font color="Maroon" title="c13">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E14:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E11:82"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="connsp_1.html#T22" title="CONNSP_1:th.22">CONNSP_1:22</a>, <a class="ref" href="jordan6.html#T10" title="JORDAN6:th.10">JORDAN6:10</a></span>;<br/>
<a NAME="E15:82"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<font color="Maroon" title="c13">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="jordan6.html#T11" title="JORDAN6:th.11">JORDAN6:11</a></span>;<br/>
<a NAME="E16:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
;<br/>
<a NAME="E17:82"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="txt" href="#E15:82"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="ref" href="jordan5c.html#D1" title="JORDAN5C:def.1">JORDAN5C:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>) or <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A14</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:82"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:82_1_1"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="jordan.html#T45" target="_self" title="JORDAN:th.45">Th45</a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E68">A14</font></span></a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_2"/><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">P</font> = <font color="Maroon" title="c13">P</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:82_1_2"/><span class="lab"><font color="Green" title="E69">A16</font></span>: 
<font color="Maroon" title="c20">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:82_1_2"/><span class="lab"><font color="Green" title="E70">A17</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font>
 <span class="kw">and </span><br/><a NAME="E6:82_1_2"/><span class="lab"><font color="Green" title="E71">A18</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E72">A19</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c19">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E8:82_1_2"/>
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:82"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">na</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:82_1_2"/><span class="lab"><font color="Green" title="E73">A20</font></span>: 
<font color="Maroon" title="c21">na</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">and </span><br/><a NAME="E11:82_1_2"/><span class="lab"><font color="Green" title="E74">A21</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">na</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">na</font> = <font color="Maroon" title="c21">na</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E13:82_1_2"/><span class="lab"><font color="Green" title="E75">A22</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">na</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E14:82_1_2"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
<font color="Maroon" title="c22">na</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:82_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E15:82_1_2"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T2" title="JORDAN16:th.2">JORDAN16:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">xa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:82_1_2"/><span class="lab"><font color="Green" title="E78">A25</font></span>: 
<font color="Maroon" title="c23">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">and </span><br/><a NAME="E18:82_1_2"/><span class="lab"><font color="Green" title="E79">A26</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E7:82_1_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">xa</font> = <font color="Maroon" title="c23">xa</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:82_1_2"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E20:82_1_2"/><span class="lab"><font color="Green" title="E80">A27</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">xa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:82_1_2"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E21:82_1_2"/><span class="lab"><font color="Green" title="E81">A28</font></span>: 
<font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:82_1_2"><span class="lab"><font color="Green" title="E78">A25</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E22:82_1_2"/><span class="lab"><font color="Green" title="E82">A29</font></span>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>) <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="txt" href="#E15:82_1_2"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="ref" href="jordan16.html#T24" title="JORDAN16:th.24">JORDAN16:24</a></span>;<br/><a NAME="E23:82_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</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="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/><a NAME="E24:82_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A30</font></span>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E25:82_1_2"/>
<font color="Maroon" title="c18">x</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="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:82_1_2"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/><a NAME="E26:82_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E27:82_1_2"/><span class="kw">then </span>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:82_1_2"><span class="lab"><font color="Green" title="E82">A29</font></span></a>, <a class="txt" href="#E24:82_1_2"><span class="lab"><font color="Green" title="E83">A30</font></span></a>, <a class="ref" href="connsp_1.html#T22" title="CONNSP_1:th.22">CONNSP_1:22</a>, <a class="ref" href="jordan6.html#T10" title="JORDAN6:th.10">JORDAN6:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E29:82_1_2"/><span class="lab"><font color="Green" title="E84">A31</font></span>: 
<font color="Maroon" title="c25">z</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="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>)
 <span class="kw">and </span><br/><a NAME="E30:82_1_2"/><span class="lab"><font color="Green" title="E85">A32</font></span>: 
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:82"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <font color="Maroon" title="c25">z</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:82_1_2"><span class="lab"><font color="Green" title="E84">A31</font></span></a></span>;<br/><a NAME="E32:82_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c11">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E33:82_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A33</font></span>: 
 ex <font color="Olive" title="b1">zz</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 <br/>( <font color="Olive" title="b1">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">z</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Olive" title="b1">zz</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">zz</font>,<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:82_1_2"><span class="lab"><font color="Green" title="E84">A31</font></span></a></span>;<br/><a NAME="E34:82_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c18">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan16.html#T2" title="JORDAN16:th.2">JORDAN16:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">za</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E36:82_1_2"/><span class="lab"><font color="Green" title="E87">A34</font></span>: 
<font color="Maroon" title="c27">za</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">and </span><br/><a NAME="E37:82_1_2"/><span class="lab"><font color="Green" title="E88">A35</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">za</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="txt" href="#E29:82_1_2"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">za</font> = <font color="Maroon" title="c27">za</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:82_1_2"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E39:82_1_2"/><span class="lab"><font color="Green" title="E89">A36</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">za</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:82_1_2"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E40:82_1_2"/><span class="lab"><font color="Green" title="E90">A37</font></span>: 
<font color="Maroon" title="c28">za</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:82_1_2"><span class="lab"><font color="Green" title="E87">A34</font></span></a>, <a class="ref" href="borsuk_1.html#T43" title="BORSUK_1:th.43">BORSUK_1:43</a></span>;<br/><a NAME="E41:82_1_2"/><span class="lab"><font color="Green" title="E91">A38</font></span>: 
<font color="Maroon" title="c22">na</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">za</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E14:82"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="txt" href="#E16:82"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E6:82_1_2"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E11:82_1_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E14:82_1_2"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E30:82_1_2"><span class="lab"><font color="Green" title="E85">A32</font></span></a>, <a class="txt" href="#E37:82_1_2"><span class="lab"><font color="Green" title="E88">A35</font></span></a>, <a class="txt" href="#E39:82_1_2"><span class="lab"><font color="Green" title="E89">A36</font></span></a>, <a class="ref" href="jordan5c.html#D1" title="JORDAN5C:def.1">JORDAN5C:def 1</a></span>;<br/><a NAME="E42:82_1_2"/><span class="lab"><font color="Green" title="E92">A39</font></span>: 
<font color="Maroon" title="c28">za</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">xa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E6:82_1_2"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E18:82_1_2"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="txt" href="#E20:82_1_2"><span class="lab"><font color="Green" title="E80">A27</font></span></a>, <a class="txt" href="#E21:82_1_2"><span class="lab"><font color="Green" title="E81">A28</font></span></a>, <a class="txt" href="#E33:82_1_2"><span class="lab"><font color="Green" title="E86">A33</font></span></a>, <a class="txt" href="#E37:82_1_2"><span class="lab"><font color="Green" title="E88">A35</font></span></a>, <a class="txt" href="#E40:82_1_2"><span class="lab"><font color="Green" title="E90">A37</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><a NAME="E43:82_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> (<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<font color="Maroon" title="c11">p1</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c11">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>, <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>),<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E44:82_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xx</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 <br/>( <font color="Olive" title="b1">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c11">p1</font>,<font color="Olive" title="b1">xx</font>,<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">xx</font>, <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> (<font color="Maroon" title="c13">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font>,<span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>)</span>)</span>),<font color="Maroon" title="c19">P</font>,<font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A6</font></span></a></span>;<br/><a NAME="E45:82_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">na</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E5:82_1_2"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E6:82_1_2"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E11:82_1_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E13:82_1_2"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E14:82_1_2"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E18:82_1_2"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="txt" href="#E21:82_1_2"><span class="lab"><font color="Green" title="E81">A28</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/><a NAME="E46:82_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">xa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">na</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="txt" href="#E11:82_1_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E18:82_1_2"><span class="lab"><font color="Green" title="E79">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E47:82_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:82_1_2"><span class="lab"><font color="Green" title="E91">A38</font></span></a>, <a class="txt" href="#E42:82_1_2"><span class="lab"><font color="Green" title="E92">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>

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