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<span class="kw">let </span><font color="Maroon" title="c1">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" 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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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 <font color="Maroon" title="c1">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> &amp; <font color="Olive" title="b3">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b4">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">q2</font> &amp;  not <!--HUMANRECFAILED-->(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b4">q2</font>,<font color="Olive" title="b3">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> ) holds <br/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b4">q2</font>,<font color="Olive" title="b3">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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"> ( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font> &amp;  not <!--HUMANRECFAILED-->(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) implies (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="E13">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</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="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) or (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c1">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="#E2:17"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<a NAME="E6:17"/>
( (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) or (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">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="c6">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c7">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="E3:17_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
( <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c7">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="c3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="topreal1.html#D1" onmouseover="rs('topreal1/D1')" onmouseout="rh()">TOPREAL1:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A7</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">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="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c6">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:17_1"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
<font color="Maroon" title="c8">x</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="c7">f</font>
 <span class="kw">and </span><br/><a NAME="E7:17_1"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:17_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c9">y</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="c7">f</font>
 <span class="kw">and </span><br/><a NAME="E10:17_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="tops_2.html#D5" onmouseover="rs('tops_2/D5')" onmouseout="rh()">TOPS_2:def 5</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">s2</font> = <font color="Maroon" title="c9">y</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_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E9:17_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a></span>;<br/>
<a NAME="E13:17_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E14:17_1"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c11">s2</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="#E9:17_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E15:17_1"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<a NAME="E16:17_1"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c10">s1</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="#E6:17_1"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<span class="kw">assume </span><a NAME="E17:17_1"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> iff  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_3"/>
( <font color="Maroon" title="c10">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s2</font> or <font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s2</font> or <font color="Maroon" title="c10">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">s2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_1"/>
<font color="Maroon" title="c10">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s2</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/><div class="add"><span class="kw">hence </span><a NAME="E2:17_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E13:17_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E16:17_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E14:17_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="jordan5c.html#T8" target="_self" onmouseover="rs('jordan5c/T8')" onmouseout="rh()">Th8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_2"/>
<font color="Maroon" title="c10">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s2</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/><div class="add"><span class="kw">hence </span><a NAME="E2:17_1_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E23">A11</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_3"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_3"/>
<font color="Maroon" title="c10">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">s2</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/><div class="add"><span class="kw">hence </span><a NAME="E2:17_1_3_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E16:17_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E15:17_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E14:17_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="jordan5c.html#T8" target="_self" onmouseover="rs('jordan5c/T8')" onmouseout="rh()">Th8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:17"/>
( (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) or (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
