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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p4</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 <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="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Olive" title="b1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Olive" title="b1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>,<font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">p1</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>  &amp; <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">p2</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>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">p3</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>  &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">p4</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>  holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font>, <font color="Olive" title="b5">q3</font>, <font color="Olive" title="b6">q4</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="b2">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; (  for <font color="Olive" title="b7">q</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> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">q</font></span>)</span></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="Olive" title="b7">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Olive" title="b5">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Olive" title="b6">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> &amp; <font color="Olive" title="b3">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b6">q4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b6">q4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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="b1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b4">q2</font>,<font color="Olive" title="b5">q3</font>,<font color="Olive" title="b1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b5">q3</font>,<font color="Olive" title="b6">q4</font>,<font color="Olive" title="b1">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c5">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="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">p1</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>  &amp; <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">p2</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>  &amp; <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">p3</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>  &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">p4</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>  implies  ex <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">q3</font>, <font color="Olive" title="b5">q4</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">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; (  for <font color="Olive" title="b6">q</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> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</font></span>)</span></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="Olive" title="b6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Olive" title="b4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Olive" title="b5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b4">q3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b4">q3</font>,<font color="Olive" title="b5">q4</font>,<font color="Maroon" title="c5">P</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c5">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="E2:62"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E3:62"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E4:62"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E5:62"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c1">p1</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">and </span><br/><a NAME="E7:62"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:62"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
<font color="Maroon" title="c2">p2</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">and </span><br/><a NAME="E9:62"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
<font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E10:62"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c3">p3</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">and </span><br/><a NAME="E11:62"/><span class="lab"><font color="Green" title="E68">A11</font></span>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E12:62"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
<font color="Maroon" title="c4">p4</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> 
 ; <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">  ex <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">q3</font>, <font color="Olive" title="b5">q4</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">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; (  for <font color="Olive" title="b6">q</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> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</font></span>)</span></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="Olive" title="b6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Olive" title="b4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Olive" title="b5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b4">q3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b4">q3</font>,<font color="Olive" title="b5">q4</font>,<font color="Maroon" title="c5">P</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E14:62"/><span class="lab"><font color="Green" title="E70">A13</font></span>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">and </span><br/><a NAME="E15:62"/><span class="lab"><font color="Green" title="E71">A14</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="xreal_1.html#T5" onmouseover="rs('xreal_1/T5')" onmouseout="rh()">XREAL_1:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">r1</font> = <font color="Maroon" title="c6">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s</font> =  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>;<br/>
<a NAME="E17:62"/><span class="lab"><font color="Green" title="E72">A15</font></span>: 
<font color="Maroon" title="c5">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="#E1:62"><span class="lab"><font color="Green" title="E58">A1</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="E18:62"/><span class="kw">then </span>
<font color="Maroon" title="c4">p4</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="jordan7.html#T5" onmouseover="rs('jordan7/T5')" onmouseout="rh()">JORDAN7:5</a></span>;<br/>
<a NAME="E19:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A16</font></span>: 
 ex <font color="Olive" title="b1">p</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> &amp; <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E20:62"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_5.html#T1" target="_self" onmouseover="rs('jgraph_5/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E21:62"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:62"/><span class="kw">then </span>
<font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 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="#E15:62"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="square_1.html#T49" onmouseover="rs('square_1/T49')" onmouseout="rh()">SQUARE_1:49</a></span>;<br/>
<a NAME="E23:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A17</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <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="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E24:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A18</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <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="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E25:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A19</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></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"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></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="#E15:62"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="square_1.html#T12" onmouseover="rs('square_1/T12')" onmouseout="rh()">SQUARE_1:12</a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E26:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A20</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>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T52" onmouseover="rs('square_1/T52')" onmouseout="rh()">SQUARE_1:52</a></span>;<br/>
<a NAME="E27:62"/><span class="lab"><font color="Green" title="E78">A21</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:62"><span class="lab"><font color="Green" title="E76">A19</font></span></a>, <a class="ref" href="square_1.html#T52" onmouseover="rs('square_1/T52')" onmouseout="rh()">SQUARE_1:52</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E29:62"/><span class="lab"><font color="Green" title="E79">A22</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <a href="jgraph_4.html#K3" title="JGRAPH_4:func.3">-FanMorphW</a> 
 <span class="kw">and </span><br/><a NAME="E30:62"/><span class="lab"><font color="Green" title="E80">A23</font></span>: 
<font color="Maroon" title="c9">f1</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="ref" href="jgraph_4.html#T41" onmouseover="rs('jgraph_4/T41')" onmouseout="rh()">JGRAPH_4:41</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q11</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">q22</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">q33</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">q44</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>;<br/>
<a NAME="E31:62"/><span class="lab"><font color="Green" title="E81">A24</font></span>: 
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <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="#E23:62"><span class="lab"><font color="Green" title="E74">A17</font></span></a>, <a class="ref" href="square_1.html#D2" onmouseover="rs('square_1/D2')" onmouseout="rh()">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E32:62"/>
<font color="Maroon" title="c3">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="jordan7.html#T5" onmouseover="rs('jordan7/T5')" onmouseout="rh()">JORDAN7:5</a></span>;<br/>
<a NAME="E33:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A25</font></span>: 
 ex <font color="Olive" title="b1">p33</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">p33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p33</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="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E34:62"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E35:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A26</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E27:62"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T46" onmouseover="rs('jgraph_4/T46')" onmouseout="rh()">JGRAPH_4:46</a></span>;<br/>
<a NAME="E36:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</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>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">r1</font> <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="#E14:62"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E15:62"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="square_1.html#T44" onmouseover="rs('square_1/T44')" onmouseout="rh()">SQUARE_1:44</a></span>;<br/>
<a NAME="E37:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A27</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <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="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/>
<a NAME="E38:62"/><span class="lab"><font color="Green" title="E85">A28</font></span>: 
( <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E12:62"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E39:62"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E40:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <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="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<a NAME="E41:62"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E42:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A29</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <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="#E37:62"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E24:62"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E43:62"/>
( <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E44:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A30</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:62"><span class="lab"><font color="Green" title="E85">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E45:62"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E46:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <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="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<a NAME="E47:62"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E48:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A31</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <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="#E37:62"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E24:62"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E49:62"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E50:62"/><span class="kw">then </span>
<font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:62"><span class="lab"><font color="Green" title="E87">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E51:62"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E52:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <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="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="ref" href="square_1.html#T15" onmouseover="rs('square_1/T15')" onmouseout="rh()">SQUARE_1:15</a></span>;<br/>
<a NAME="E53:62"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/>
<a NAME="E54:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A32</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span> <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="#E37:62"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E24:62"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E55:62"/>
1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</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">p3</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="#E33:62"><span class="lab"><font color="Green" title="E82">A25</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="E56:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:62"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="txt" href="#E42:62"><span class="lab"><font color="Green" title="E86">A29</font></span></a>, <a class="ref" href="square_1.html#T48" onmouseover="rs('square_1/T48')" onmouseout="rh()">SQUARE_1:48</a></span>;<br/>
<a NAME="E57:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E58:62"/>
<font color="Maroon" title="c2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="jordan7.html#T5" onmouseover="rs('jordan7/T5')" onmouseout="rh()">JORDAN7:5</a></span>;<br/>
<a NAME="E59:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A35</font></span>: 
 ex <font color="Olive" title="b1">p22</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">p22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p22</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="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E60:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A36</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></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="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T33" onmouseover="rs('jgraph_4/T33')" onmouseout="rh()">JGRAPH_4:33</a></span>;<br/>
<a NAME="E61:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A37</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E62:62"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E59:62"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E33:62"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E63:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A38</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E27:62"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T46" onmouseover="rs('jgraph_4/T46')" onmouseout="rh()">JGRAPH_4:46</a></span>;<br/>
<a NAME="E64:62"/><span class="lab"><font color="Green" title="E96">A39</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span></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="#E33:62"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T33" onmouseover="rs('jgraph_4/T33')" onmouseout="rh()">JGRAPH_4:33</a></span>;<br/>
<a NAME="E65:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A40</font></span>: 
<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E66:62"/>
1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</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">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="#E59:62"><span class="lab"><font color="Green" title="E92">A35</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="E67:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A41</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:62"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="txt" href="#E48:62"><span class="lab"><font color="Green" title="E88">A31</font></span></a>, <a class="ref" href="square_1.html#T48" onmouseover="rs('square_1/T48')" onmouseout="rh()">SQUARE_1:48</a></span>;<br/>
<a NAME="E68:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A42</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E69:62"/><span class="lab"><font color="Green" title="E100">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E67:62"><span class="lab"><font color="Green" title="E98">A41</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E70:62"/>
1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p4</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="c4">p4</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="#E19:62"><span class="lab"><font color="Green" title="E73">A16</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="E71:62"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:62"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="txt" href="#E19:62"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E24:62"><span class="lab"><font color="Green" title="E75">A18</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="square_1.html#T48" onmouseover="rs('square_1/T48')" onmouseout="rh()">SQUARE_1:48</a></span>;<br/>
<a NAME="E72:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A44</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E11:62"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E73:62"/>
<font color="Maroon" title="c1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E17:62"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="ref" href="jordan7.html#T5" onmouseover="rs('jordan7/T5')" onmouseout="rh()">JORDAN7:5</a></span>;<br/>
<a NAME="E74:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A45</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="c1">p1</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="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E75:62"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c1">p1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">p2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E59:62"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="ref" href="jgraph_5.html#T46" target="_self" onmouseover="rs('jgraph_5/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E76:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A46</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E27:62"><span class="lab"><font color="Green" title="E78">A21</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T46" onmouseover="rs('jgraph_4/T46')" onmouseout="rh()">JGRAPH_4:46</a></span>;<br/>
<a NAME="E77:62"/>
1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:62"><span class="lab"><font color="Green" title="E102">A45</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="E78:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A47</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c1">p1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:62"><span class="lab"><font color="Green" title="E102">A45</font></span></a>, <a class="txt" href="#E31:62"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="txt" href="#E54:62"><span class="lab"><font color="Green" title="E89">A32</font></span></a>, <a class="ref" href="square_1.html#T48" onmouseover="rs('square_1/T48')" onmouseout="rh()">SQUARE_1:48</a></span>;<br/>
<a NAME="E79:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E5:62"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E80:62"/><span class="lab"><font color="Green" title="E106">A49</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span></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="#E74:62"><span class="lab"><font color="Green" title="E102">A45</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T33" onmouseover="rs('jgraph_4/T33')" onmouseout="rh()">JGRAPH_4:33</a></span>;<br/>
<a NAME="E81:62"/><span class="kw">then </span>
<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E82:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A50</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E80:62"><span class="lab"><font color="Green" title="E106">A49</font></span></a>, <a class="txt" href="#E60:62"><span class="lab"><font color="Green" title="E93">A36</font></span></a>, <a class="txt" href="#E61:62"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E79:62"><span class="lab"><font color="Green" title="E105">A48</font></span></a>, <a class="txt" href="#E69:62"><span class="lab"><font color="Green" title="E100">A43</font></span></a>, <a class="txt" href="#E68:62"><span class="lab"><font color="Green" title="E99">A42</font></span></a>, <a class="txt" href="#E76:62"><span class="lab"><font color="Green" title="E103">A46</font></span></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" onmouseover="rs('jgraph_5/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E83:62"/><span class="lab"><font color="Green" title="E108">A51</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E7:62"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E67:62"><span class="lab"><font color="Green" title="E98">A41</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E84:62"/><span class="lab"><font color="Green" title="E109">A52</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E5:62"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E78:62"><span class="lab"><font color="Green" title="E104">A47</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E85:62"/><span class="lab"><font color="Green" title="E110">A53</font></span>: 
 for <font color="Olive" title="b1">q</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> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">q</font></span>)</span></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="Olive" title="b1">q</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="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T33" onmouseover="rs('jgraph_4/T33')" onmouseout="rh()">JGRAPH_4:33</a></span>;<br/>
<a NAME="E86:62"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E56:62"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E87:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A54</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E60:62"><span class="lab"><font color="Green" title="E93">A36</font></span></a>, <a class="txt" href="#E61:62"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E64:62"><span class="lab"><font color="Green" title="E96">A39</font></span></a>, <a class="txt" href="#E65:62"><span class="lab"><font color="Green" title="E97">A40</font></span></a>, <a class="txt" href="#E69:62"><span class="lab"><font color="Green" title="E100">A43</font></span></a>, <a class="txt" href="#E63:62"><span class="lab"><font color="Green" title="E95">A38</font></span></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" onmouseover="rs('jgraph_5/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E88:62"/><span class="lab"><font color="Green" title="E112">A55</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="#E9:62"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="txt" href="#E26:62"><span class="lab"><font color="Green" title="E77">A20</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="txt" href="#E56:62"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="jgraph_4.html#T43" onmouseover="rs('jgraph_4/T43')" onmouseout="rh()">JGRAPH_4:43</a></span>;<br/>
<a NAME="E89:62"/><span class="lab"><font color="Green" title="E113">A56</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span></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="#E19:62"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E29:62"><span class="lab"><font color="Green" title="E79">A22</font></span></a>, <a class="ref" href="jgraph_4.html#T33" onmouseover="rs('jgraph_4/T33')" onmouseout="rh()">JGRAPH_4:33</a></span>;<br/>
<a NAME="E90:62"/><span class="kw">then </span>
<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<a NAME="E91:62"/><span class="kw">then </span>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c9">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E64:62"><span class="lab"><font color="Green" title="E96">A39</font></span></a>, <a class="txt" href="#E65:62"><span class="lab"><font color="Green" title="E97">A40</font></span></a>, <a class="txt" href="#E89:62"><span class="lab"><font color="Green" title="E113">A56</font></span></a>, <a class="txt" href="#E57:62"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="txt" href="#E72:62"><span class="lab"><font color="Green" title="E101">A44</font></span></a>, <a class="txt" href="#E35:62"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="ref" href="jgraph_5.html#T51" target="_self" onmouseover="rs('jgraph_5/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E92:62"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font>, <font color="Olive" title="b4">q3</font>, <font color="Olive" title="b5">q4</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">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; (  for <font color="Olive" title="b6">q</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> holds  <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</font></span>)</span></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="Olive" title="b6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; <font color="Olive" title="b2">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> &amp; <font color="Olive" title="b4">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> &amp; <font color="Olive" title="b5">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b3">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b4">q3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b5">q4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b3">q2</font>,<font color="Olive" title="b4">q3</font>,<font color="Maroon" title="c5">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b4">q3</font>,<font color="Olive" title="b5">q4</font>,<font color="Maroon" title="c5">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:62"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E85:62"><span class="lab"><font color="Green" title="E110">A53</font></span></a>, <a class="txt" href="#E84:62"><span class="lab"><font color="Green" title="E109">A52</font></span></a>, <a class="txt" href="#E83:62"><span class="lab"><font color="Green" title="E108">A51</font></span></a>, <a class="txt" href="#E57:62"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="txt" href="#E88:62"><span class="lab"><font color="Green" title="E112">A55</font></span></a>, <a class="txt" href="#E72:62"><span class="lab"><font color="Green" title="E101">A44</font></span></a>, <a class="txt" href="#E82:62"><span class="lab"><font color="Green" title="E107">A50</font></span></a>, <a class="txt" href="#E87:62"><span class="lab"><font color="Green" title="E111">A54</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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