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<span class="kw">let </span><font color="Maroon" title="c1">cn</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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><br/>  for <font color="Olive" title="b5">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>  st  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b6">p</font> where <font color="Olive" title="b6">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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b6">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span> } </span>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p2</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b5">P</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">  for <font color="Olive" title="b1">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>  st  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b2">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span> } </span>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p2</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>; <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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c6">P</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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span> } </span>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p2</font> implies  <a href="jordan6.html#R1" title="JORDAN6: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> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">cn</font> &amp; <font color="Maroon" title="c1">cn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c6">P</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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">and </span><br/><a NAME="E4:61"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">and </span><br/><a NAME="E5:61"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <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">  <a href="jordan6.html#R1" title="JORDAN6: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></span><br/>

<a NAME="E6:61"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c6">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_3.html#T26" onmouseover="rs('jgraph_3/T26')" onmouseout="rh()">JGRAPH_3:26</a></span>;<br/>
<a NAME="E7:61"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_5.html#T29" target="_self" onmouseover="rs('jgraph_5/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E8:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/>
<a NAME="E9:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_4.html#T113" onmouseover="rs('jgraph_4/T113')" onmouseout="rh()">JGRAPH_4:113</a></span>;<br/>
<a NAME="E10:61"/>
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E11:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A9</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:61"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E13:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A10</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:61"/><span class="lab"><font color="Green" title="E67">A11</font></span>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" onmouseover="rs('jgraph_5/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<a NAME="E15:61"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
<font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="jgraph_4.html#T133" onmouseover="rs('jgraph_4/T133')" onmouseout="rh()">JGRAPH_4:133</a></span>;<br/>
<a NAME="E16:61"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" onmouseover="rs('jgraph_5/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E70">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><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="E1:61_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> or <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E59">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><span class="kw">case </span></a><a NAME="E1:61_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61"><span class="lab"><font color="Green" title="E69">A13</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="61_1_1_2"><span class="kw">case </span></a><a NAME="E1:61_1_1_2"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_1_1_2"/>
<font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61"><span class="lab"><font color="Green" title="E67">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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E71">A15</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">now </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"> (  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E72">A16</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span><br/><a NAME="E2:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E13:61"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E3:61_2"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E17:61"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="ref" href="jordan7.html#T3" onmouseover="rs('jordan7/T3')" onmouseout="rh()">JORDAN7:3</a></span>;<br/><a NAME="E4:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A17</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="jgraph_3.html#T26" onmouseover="rs('jgraph_3/T26')" onmouseout="rh()">JGRAPH_3:26</a>, <a class="ref" href="jordan6.html#T57" onmouseover="rs('jordan6/T57')" onmouseout="rh()">JORDAN6:57</a></span>;<br/><a NAME="E5:61_2"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="jordan7.html#T1" onmouseover="rs('jordan7/T1')" onmouseout="rh()">JORDAN7:1</a></span>;<br/><a NAME="E6:61_2"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E14:61"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="txt" href="#E4:61_2"><span class="lab"><font color="Green" title="E73">A17</font></span></a>, <a class="ref" href="jgraph_3.html#T26" onmouseover="rs('jgraph_3/T26')" onmouseout="rh()">JGRAPH_3:26</a>, <a class="ref" href="jordan6.html#T56" onmouseover="rs('jordan6/T56')" onmouseout="rh()">JORDAN6:56</a></span>;<br/><span class="kw">hence </span><a NAME="E7:61_2"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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><span class="kw">end;</span></div>
<a NAME="E19:61"/><span class="lab"><font color="Green" title="E72">A18</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</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> : ( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_5.html#T34" target="_self" onmouseover="rs('jgraph_5/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E20:61"/><span class="lab"><font color="Green" title="E73">A19</font></span>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</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> : ( <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="jgraph_5.html#T35" target="_self" onmouseover="rs('jgraph_5/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_3"/>
( ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E59">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1"><span class="kw">suppose </span></a><a NAME="E1:61_3_1"/><span class="lab"><font color="Green" title="E74">A20</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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="jordan6.html#R1" title="JORDAN6: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></span><br/><div class="add"><a NAME="E2:61_3_1"/><span class="lab"><font color="Green" title="E75">A21</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="jgraph_4.html#T128" onmouseover="rs('jgraph_4/T128')" onmouseout="rh()">JGRAPH_4:128</a></span>;<br/><a NAME="E3:61_3_1"/><span class="lab"><font color="Green" title="E76">A22</font></span>: 
 ex <font color="Olive" title="b1">p9</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">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="txt" href="#E1:61_3_1"><span class="lab"><font color="Green" title="E74">A20</font></span></a></span>;<br/><a NAME="E4:61_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p10</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">p10</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p10</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/><a NAME="E5:61_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A23</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E2:61_3_1"><span class="lab"><font color="Green" title="E75">A21</font></span></a></span>;<br/><a NAME="E6:61_3_1"/><span class="lab"><font color="Green" title="E78">A24</font></span>: 
 ex <font color="Olive" title="b1">p8</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">p8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Olive" title="b1">p8</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="txt" href="#E1:61_3_1"><span class="lab"><font color="Green" title="E74">A20</font></span></a></span>;<br/><a NAME="E7:61_3_1"/>
<font color="Maroon" title="c5">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E3:61_3_1"><span class="lab"><font color="Green" title="E76">A22</font></span></a>, <a class="ref" href="jgraph_5.html#T57" target="_self" onmouseover="rs('jgraph_5/T57')" onmouseout="rh()">Th57</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_1"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="txt" href="#E18:61"><span class="lab"><font color="Green" title="E71">A15</font></span></a>, <a class="txt" href="#E1:61_3_1"><span class="lab"><font color="Green" title="E74">A20</font></span></a>, <a class="txt" href="#E6:61_3_1"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="txt" href="#E5:61_3_1"><span class="lab"><font color="Green" title="E77">A23</font></span></a>, <a class="ref" href="jgraph_4.html#T113" onmouseover="rs('jgraph_4/T113')" onmouseout="rh()">JGRAPH_4:113</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="jordan6.html#D10" onmouseover="rs('jordan6/D10')" onmouseout="rh()">JORDAN6:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_2"><span class="kw">suppose </span></a><a NAME="E1:61_3_2"/><span class="lab"><font color="Green" title="E74">A25</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</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="jordan6.html#R1" title="JORDAN6: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></span><br/><div class="add"><a NAME="E2:61_3_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p8</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">p8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Olive" title="b1">p8</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E72">A18</font></span></a></span>;<br/><a NAME="E3:61_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_4.html#T113" onmouseover="rs('jgraph_4/T113')" onmouseout="rh()">JGRAPH_4:113</a></span>;<br/><a NAME="E4:61_3_2"/>
 ex <font color="Olive" title="b1">p9</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">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="txt" href="#E1:61_3_2"><span class="lab"><font color="Green" title="E74">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_3_2"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E1:61_3_2"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E3:61_3_2"><span class="lab"><font color="Green" title="E75">A26</font></span></a>, <a class="ref" href="jgraph_4.html#T113" onmouseover="rs('jgraph_4/T113')" onmouseout="rh()">JGRAPH_4:113</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="jordan6.html#D10" onmouseover="rs('jordan6/D10')" onmouseout="rh()">JORDAN6:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3"><span class="kw">suppose </span></a><a NAME="E1:61_3_3"/><span class="lab"><font color="Green" title="E74">A27</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c2">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</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="jordan6.html#R1" title="JORDAN6: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></span><br/><div class="add"><a NAME="E2:61_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A28</font></span>: 
 ex <font color="Olive" title="b1">p8</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">p8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Olive" title="b1">p8</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E73">A19</font></span></a></span>;<br/><a NAME="E3:61_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A29</font></span>: 
 ex <font color="Olive" title="b1">p10</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">p10</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p10</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/><a NAME="E4:61_3_3"/><span class="lab"><font color="Green" title="E77">A30</font></span>: 
 ex <font color="Olive" title="b1">p9</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">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">p9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61"><span class="lab"><font color="Green" title="E73">A19</font></span></a>, <a class="txt" href="#E1:61_3_3"><span class="lab"><font color="Green" title="E74">A27</font></span></a></span>;<br/><a NAME="E5:61_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A31</font></span>: 
 ex <font color="Olive" title="b1">p11</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">p11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p11</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/><a NAME="E6:61_3_3"/><span class="lab"><font color="Green" title="E79">A32</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E4:61_3_3"><span class="lab"><font color="Green" title="E77">A30</font></span></a>, <a class="ref" href="jgraph_5.html#T57" target="_self" onmouseover="rs('jgraph_5/T57')" onmouseout="rh()">Th57</a></span>;<br/><a NAME="E7:61_3_3"/><span class="lab"><font color="Green" title="E80">A33</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="jgraph_4.html#T128" onmouseover="rs('jgraph_4/T128')" onmouseout="rh()">JGRAPH_4:128</a></span>;<br/><a NAME="E8:61_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A34</font></span>: 
<font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E5:61_3_3"><span class="lab"><font color="Green" title="E78">A31</font></span></a></span>;<br/><a NAME="E9:61_3_3"/><span class="lab"><font color="Green" title="E82">A35</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E2:61_3_3"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="ref" href="jgraph_5.html#T57" target="_self" onmouseover="rs('jgraph_5/T57')" onmouseout="rh()">Th57</a></span>;<br/><a NAME="E10:61_3_3"/><span class="lab"><font color="Green" title="E83">A36</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">q1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="jgraph_4.html#T128" onmouseover="rs('jgraph_4/T128')" onmouseout="rh()">JGRAPH_4:128</a></span>;<br/><a NAME="E11:61_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A37</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:61_3_3"><span class="lab"><font color="Green" title="E76">A29</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_3_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp; ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) ) ) or ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp; ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) ) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1"><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="E1:61_3_3_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:61_3_3_1_1_1"/><span class="lab"><font color="Green" title="E85">A38</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span><br/><div class="add"><a NAME="E2:61_3_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">cn</font> <a href="jgraph_4.html#K9" title="JGRAPH_4:func.9">-FanMorphS</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="jgraph_4.html#T113" onmouseover="rs('jgraph_4/T113')" onmouseout="rh()">JGRAPH_4:113</a></span>;<br/><a NAME="E3:61_3_3_1_1_1"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E8:61_3_3"><span class="lab"><font color="Green" title="E81">A34</font></span></a>, <a class="txt" href="#E1:61_3_3_1_1_1"><span class="lab"><font color="Green" title="E85">A38</font></span></a>, <a class="ref" href="jordan7.html#T3" onmouseover="rs('jordan7/T3')" onmouseout="rh()">JORDAN7:3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_3_3_1_1_1"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:61_3_3_1_1_2"/><span class="lab"><font color="Green" title="E85">A39</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_3_3_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) ) ) or ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) ) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1_2_1_1"><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="E1:61_3_3_1_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E2:61_3_3"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="txt" href="#E1:61_3_3_1_1_2"><span class="lab"><font color="Green" title="E85">A39</font></span></a>, <a class="ref" href="jgraph_5.html#T48" target="_self" onmouseover="rs('jgraph_5/T48')" onmouseout="rh()">Th48</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:61_3_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E86">A40</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span><br/><div class="add"><a NAME="E2:61_3_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A41</font></span>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E88">A42</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_3_3_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> implies <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:61_3_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E89">A43</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font></span><br/><a NAME="E2:61_3_3_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_3_3"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="txt" href="#E4:61_3_3"><span class="lab"><font color="Green" title="E77">A30</font></span></a>, <a class="ref" href="xreal_1.html#T58" onmouseover="rs('xreal_1/T58')" onmouseout="rh()">XREAL_1:58</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_3_3_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <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="#E1:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A40</font></span></a>, <a class="txt" href="#E2:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A41</font></span></a>, <a class="txt" href="#E1:61_3_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E89">A43</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</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><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> = 
1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_3_3"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="ref" href="jgraph_3.html#T1" onmouseover="rs('jgraph_3/T1')" onmouseout="rh()">JGRAPH_3:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_3_3"><span class="lab"><font color="Green" title="E78">A31</font></span></a>, <a class="txt" href="#E1:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A40</font></span></a>, <a class="ref" href="jgraph_3.html#T1" onmouseover="rs('jgraph_3/T1')" onmouseout="rh()">JGRAPH_3:1</a></span>
;<br/><a NAME="E5:61_3_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A44</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T40" onmouseover="rs('square_1/T40')" onmouseout="rh()">SQUARE_1:40</a></span>;<br/><a NAME="E6:61_3_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E7:61_3_3_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E8:61_3_3"><span class="lab"><font color="Green" title="E81">A34</font></span></a>, <a class="txt" href="#E1:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A40</font></span></a>, <a class="txt" href="#E5:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E2:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A41</font></span></a>, <a class="txt" href="#E3:61_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A42</font></span></a>, <a class="ref" href="jgraph_3.html#T26" onmouseover="rs('jgraph_3/T26')" onmouseout="rh()">JGRAPH_3:26</a>, <a class="ref" href="jordan6.html#T56" onmouseover="rs('jordan6/T56')" onmouseout="rh()">JORDAN6:56</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_3_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_3_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:61_3_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )</span><br/><div class="add"><a NAME="E2:61_3_3_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c2">p1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_3_3"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="txt" href="#E5:61_3_3"><span class="lab"><font color="Green" title="E78">A31</font></span></a></span>;<br/><a NAME="E3:61_3_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">q1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E2:61_3_3"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="txt" href="#E4:61_3_3"><span class="lab"><font color="Green" title="E77">A30</font></span></a>, <a class="txt" href="#E3:61_3_3"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="txt" href="#E5:61_3_3"><span class="lab"><font color="Green" title="E78">A31</font></span></a>, <a class="ref" href="jgraph_5.html#T27" target="_self" onmouseover="rs('jgraph_5/T27')" onmouseout="rh()">Th27</a></span>;<br/><a NAME="E4:61_3_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E13:61"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="txt" href="#E1:61_3_3"><span class="lab"><font color="Green" title="E74">A27</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E5:61_3_3_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E10:61_3_3"><span class="lab"><font color="Green" title="E83">A36</font></span></a>, <a class="txt" href="#E7:61_3_3"><span class="lab"><font color="Green" title="E80">A33</font></span></a>, <a class="txt" href="#E9:61_3_3"><span class="lab"><font color="Green" title="E82">A35</font></span></a>, <a class="txt" href="#E6:61_3_3"><span class="lab"><font color="Green" title="E79">A32</font></span></a>, <a class="txt" href="#E3:61_3_3"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="txt" href="#E5:61_3_3"><span class="lab"><font color="Green" title="E78">A31</font></span></a>, <a class="txt" href="#E11:61_3_3"><span class="lab"><font color="Green" title="E84">A37</font></span></a>, <a class="txt" href="#E8:61_3_3"><span class="lab"><font color="Green" title="E81">A34</font></span></a>, <a class="txt" href="#E3:61_3_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A45</font></span></a>, <a class="ref" href="jgraph_5.html#T56" target="_self" onmouseover="rs('jgraph_5/T56')" onmouseout="rh()">Th56</a></span>;<br/><span class="kw">hence </span><a NAME="E6:61_3_3_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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><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="E3:61_3_3_1_1_2"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</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><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="E13:61_3_3"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font> &amp;  not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> &amp;  <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>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c6">P</font>, <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c6">P</font> ) )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="jordan6.html#D10" onmouseover="rs('jordan6/D10')" onmouseout="rh()">JORDAN6:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
