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<span class="kw">set </span><font color="Maroon" title="c1">K1a</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">K0a</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">K0</font>, <font color="Maroon" title="c5">Kb</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">K0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) </span> } </span>  &amp; <font color="Maroon" title="c5">Kb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) ) </span> } </span>  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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds <br/> not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> ) &amp; (  for <font color="Olive" title="b2">r</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  not <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> ) &amp; (  for <font color="Olive" title="b2">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">K0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) </span> } </span>  &amp; <font color="Maroon" title="c5">Kb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) ) </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">  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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds <br/> not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> ) &amp; (  for <font color="Olive" title="b2">r</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  not <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> ) &amp; (  for <font color="Olive" title="b2">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T9" target="_self" onmouseover="rs('jgraph_2/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<a NAME="E4:51"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E5:51"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds <br/> not <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; <font color="Maroon" title="c7">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> implies  not <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 <span class="kw">and </span><br/><a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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">  not <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font></span><br/>

<a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
 ex <font color="Olive" title="b1">p3</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">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p3</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">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> 1 &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p3</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p3</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> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; contradiction ) or ( <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1"/>
( <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> or <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p4</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><span class="kw"> such that </span><br/><a NAME="E3:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A9</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">and </span><br/><a NAME="E4:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A11</font></span>: 
<font color="Maroon" title="c8">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> 1
 <span class="kw">and </span><br/><a NAME="E6:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A12</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E7:51_1_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A13</font></span>: 
<font color="Maroon" title="c8">p4</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; contradiction ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A14</font></span>: 
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</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="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A15</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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; contradiction ) or ( <font color="Maroon" title="c7">t</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; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  or <font color="Maroon" title="c7">t</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A16</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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"> contradiction</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A16</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A17</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A15</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A18</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A18</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E77">A19</font></span>: 
<font color="Maroon" title="c7">t</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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A15</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E77">A19</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E4:51_1_1_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <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/><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A20</font></span>: 
(  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A21</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</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; contradiction ) or ( <font color="Maroon" title="c7">t</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; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</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>  or <font color="Maroon" title="c7">t</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A22</font></span>: 
<font color="Maroon" title="c7">t</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"> contradiction</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A22</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A23</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A21</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A23</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_1_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
<font color="Maroon" title="c7">t</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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A21</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/><a NAME="E3:51_1_1_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E4:51_1_1_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <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/><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:51_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p4</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><span class="kw"> such that </span><br/><a NAME="E3:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">and </span><br/><a NAME="E4:51_1_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A26</font></span>: 
( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; contradiction ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A27</font></span>: 
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A28</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</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="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:51_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A29</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; contradiction ) or ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 &amp; contradiction ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1"/>
( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A26</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" onmouseover="rs('xcmplx_0/D7')" onmouseout="rh()">XCMPLX_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A31</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:51_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_2"/>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" onmouseover="rs('xcmplx_0/D7')" onmouseout="rh()">XCMPLX_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A33</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A33</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:51_1_1_1_1_2_1_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E75">A34</font></span>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">K01</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T17" target="_self" onmouseover="rs('jgraph_2/T17')" onmouseout="rh()">Th17</a></span>;<br/><span class="lab"><font color="Green" title="E76">A35</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><span class="lab"><font color="Green" title="E77">A36</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>

<br/>.= 
<font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E5:51_1_1_1_1_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><a NAME="E6:51_1_1_1_1_2_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A37</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A36</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E7:51_1_1_1_1_2_1_1_1_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T15" target="_self" onmouseover="rs('jgraph_2/T15')" onmouseout="rh()">Th15</a></span>;<br/><a NAME="E8:51_1_1_1_1_2_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A38</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E78">A37</font></span></a></span>;<br/><a NAME="E9:51_1_1_1_1_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><a NAME="E10:51_1_1_1_1_2_1_1_1_1_1_3"/><span class="kw">then </span>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E79">A38</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E76">A35</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><a NAME="E11:51_1_1_1_1_2_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A39</font></span>: 
 ex <font color="Olive" title="b1">p5</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">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> &amp; ( ( <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p5</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> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_3_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_3_3_1"/>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="txt" href="#E11:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E80">A39</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_3_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_3_3_1_1"/><span class="lab"><font color="Green" title="E81">A40</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3_3_1_1"><span class="lab"><font color="Green" title="E81">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_3_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_3_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_3_3_1_2"/><span class="lab"><font color="Green" title="E81">A41</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3_3_1_2"><span class="lab"><font color="Green" title="E81">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_3_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A34</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:51_1_1_1_1_2_1_1_1_1_1_3"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E75">A42</font></span>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">K01</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T17" target="_self" onmouseover="rs('jgraph_2/T17')" onmouseout="rh()">Th17</a></span>;<br/><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><a NAME="E4:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A43</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K01</font>

<br/>.= 
<font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E6:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a></span>;<br/><a NAME="E7:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A44</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E8:51_1_1_1_1_2_1_1_1_1_1_4"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T15" target="_self" onmouseover="rs('jgraph_2/T15')" onmouseout="rh()">Th15</a></span>;<br/><a NAME="E9:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A45</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A44</font></span></a></span>;<br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K01</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">K01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E11:51_1_1_1_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A46</font></span>: 
 ex <font color="Olive" title="b1">p5</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">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> &amp; ( ( <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p5</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> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E9:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A45</font></span></a>, <a class="txt" href="#E4:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E76">A43</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_4_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_4_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_4_3_1"/>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="txt" href="#E11:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E79">A46</font></span></a>, <a class="ref" href="xreal_1.html#T25" onmouseover="rs('xreal_1/T25')" onmouseout="rh()">XREAL_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_4_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_4_3_1_1"/><span class="lab"><font color="Green" title="E80">A47</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4_3_1_1"><span class="lab"><font color="Green" title="E80">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_4_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_1_1_1_4_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_1_1_1_4_3_1_2"/><span class="lab"><font color="Green" title="E80">A48</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4_3_1_2"><span class="lab"><font color="Green" title="E80">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_1_1_1_4_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E75">A42</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:51_1_1_1_1_2_1_1_1_1_1_4"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A49</font></span>: 
(  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A50</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:51_1_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A51</font></span>: 
<font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 &amp; contradiction ) or ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; contradiction ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1"/>
( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) or ( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A26</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_1_1_1_1"/><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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A52</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" onmouseover="rs('xcmplx_0/D7')" onmouseout="rh()">XCMPLX_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A49</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:51_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c8">p4</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> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A53</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A53</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" onmouseover="rs('xcmplx_0/D7')" onmouseout="rh()">XCMPLX_0:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_1_1_1_1_2_1_1_2_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A49</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:51_1_1_1_1_2_1_1_2_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E75">A54</font></span>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">K11</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T18" target="_self" onmouseover="rs('jgraph_2/T18')" onmouseout="rh()">Th18</a></span>;<br/><span class="lab"><font color="Green" title="E76">A55</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>

<br/>.= 
<font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E4:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E77">A56</font></span>: 
( ( <font color="Maroon" title="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A49</font></span></a>, <a class="ref" href="jgraph_2.html#T13" target="_self" onmouseover="rs('jgraph_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E5:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a></span>;<br/><a NAME="E6:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A57</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><a NAME="E7:51_1_1_1_1_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E4:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A56</font></span></a></span>;<br/><a NAME="E8:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A58</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E76">A55</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E9:51_1_1_1_1_2_1_1_2_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T16" target="_self" onmouseover="rs('jgraph_2/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E10:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A59</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E79">A58</font></span></a></span>;<br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E12:51_1_1_1_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A60</font></span>: 
 ex <font color="Olive" title="b1">p5</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">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> &amp; ( ( <font color="Olive" title="b1">p5</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="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p5</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E10:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="txt" href="#E6:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E78">A57</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">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> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_3_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_3_3_1"/>
( <font color="Maroon" title="c8">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> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="txt" href="#E12:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E81">A60</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_3_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_3_3_1_1"/><span class="lab"><font color="Green" title="E82">A61</font></span>: 
<font color="Maroon" title="c8">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> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3_3_1_1"><span class="lab"><font color="Green" title="E82">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_2_1_1_3_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A50</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_3_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_3_3_1_2"/><span class="lab"><font color="Green" title="E82">A62</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3_3_1_2"><span class="lab"><font color="Green" title="E82">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_2_1_1_3_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A50</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:51_1_1_1_1_2_1_1_2_1_1_3"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_4"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E75">A63</font></span>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">K11</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T18" target="_self" onmouseover="rs('jgraph_2/T18')" onmouseout="rh()">Th18</a></span>;<br/><span class="lab"><font color="Green" title="E76">A64</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/><span class="lab"><font color="Green" title="E77">A65</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">K11</font>

<br/>.= 
<font color="Maroon" title="c9">K11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><a NAME="E5:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E78">A66</font></span>: 
( ( <font color="Maroon" title="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A49</font></span></a>, <a class="ref" href="jgraph_2.html#T13" target="_self" onmouseover="rs('jgraph_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E6:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a></span>;<br/><a NAME="E7:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E8:51_1_1_1_1_2_1_1_2_1_1_4"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T16" target="_self" onmouseover="rs('jgraph_2/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E9:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A68</font></span>: 
<span class="p1">(<span class="default"><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">K11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">K11</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E79">A67</font></span></a></span>;<br/><a NAME="E10:51_1_1_1_1_2_1_1_2_1_1_4"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E5:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E78">A66</font></span></a></span>;<br/><a NAME="E11:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="kw">then </span>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E80">A68</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E76">A64</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><a NAME="E12:51_1_1_1_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A69</font></span>: 
 ex <font color="Olive" title="b1">p5</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">p5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p4</font> &amp; ( ( <font color="Olive" title="b1">p5</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="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p5</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p5</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p5</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_4_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">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> 1 &amp; contradiction ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_4_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_4_3_1"/>
( <font color="Maroon" title="c8">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> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="txt" href="#E12:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="ref" href="xreal_1.html#T25" onmouseover="rs('xreal_1/T25')" onmouseout="rh()">XREAL_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_4_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_4_3_1_1"/><span class="lab"><font color="Green" title="E82">A70</font></span>: 
<font color="Maroon" title="c8">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> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4_3_1_1"><span class="lab"><font color="Green" title="E82">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_2_1_1_4_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A50</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_1_1_2_1_1_2_1_1_4_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_1_1_1_1_2_1_1_2_1_1_4_3_1_2"/><span class="lab"><font color="Green" title="E82">A71</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A51</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4_3_1_2"><span class="lab"><font color="Green" title="E82">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><span class="kw">hence </span><a NAME="E3:51_1_1_1_1_2_1_1_2_1_1_4_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:51_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E2:51_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A50</font></span></a>, <a class="txt" href="#E1:51_1_1_1_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:51_1_1_1_1_2_1_1_2_1_1_4"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_1_1_1_1_2_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:51_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_1_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:51_1"/>
 not <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:51"/><span class="lab"><font color="Green" title="E66">A72</font></span>: 
 for <font color="Olive" title="b1">t</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  not <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> holds <br/><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">t</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"> (  not <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> implies <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E67">A73</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font></span><br/>

<a NAME="E2:51_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A74</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:51_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A75</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="jgraph_2.html#T3" target="_self" onmouseover="rs('jgraph_2/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E4:51_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:51_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:51_2"/><span class="kw">then </span>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">p4</font> = <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:51_2"/><span class="lab"><font color="Green" title="E70">A76</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E67">A73</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1"/>
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A77</font></span>: 
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E72">A78</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A79</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A80</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_1_1_1_1_1_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c7">t</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> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A81</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A80</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="E4:51_2_1_1_1_1_1_1_1_1_1"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/><a NAME="E5:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A82</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A81</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T14" onmouseover="rs('square_1/T14')" onmouseout="rh()">SQUARE_1:14</a></span>;<br/><a NAME="E7:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</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="E8:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</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="c7">t</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#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E9:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E10:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A83</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E11:51_2_1_1_1_1_1_1_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A80</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/><a NAME="E12:51_2_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A84</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</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="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E13:51_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A82</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><span class="kw">hence </span><a NAME="E14:51_2_1_1_1_1_1_1_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></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="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E12:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A84</font></span></a>, <a class="txt" href="#E10:51_2_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A83</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E74">A85</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E75">A86</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">t</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> 1 implies <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_2_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">t</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> 1
 ; <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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1</span><br/><a NAME="E2:51_2_1_1_1_1_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E74">A85</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_1_1_1_1_1_1_2_1"/><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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_2_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:51_2_1_1_1_1_1_1_1_1_2"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/><a NAME="E4:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A87</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A86</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A88</font></span>: 
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T14" onmouseover="rs('square_1/T14')" onmouseout="rh()">SQUARE_1:14</a></span>;<br/><a NAME="E6:51_2_1_1_1_1_1_1_1_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><a NAME="E7:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E77">A88</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="E8:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E9:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E10:51_2_1_1_1_1_1_1_1_1_2"/><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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">t</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="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></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="E11:51_2_1_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A89</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</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="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E12:51_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A87</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><span class="kw">hence </span><a NAME="E13:51_2_1_1_1_1_1_1_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></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="#E1:51_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A79</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E74">A85</font></span></a>, <a class="txt" href="#E11:51_2_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A89</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_2_1_1_1_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A90</font></span>: 
<font color="Maroon" title="c7">t</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>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; contradiction ) or ( <font color="Maroon" title="c7">t</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;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E73">A90</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E74">A91</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:51_2_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_2_1_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A91</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E74">A92</font></span>: 
<font color="Maroon" title="c7">t</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> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A93</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c7">t</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> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</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="E4:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A94</font></span>: 
( <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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="E5:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/><a NAME="E6:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A95</font></span>: 
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A94</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E8:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A96</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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="E9:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A95</font></span></a>, <a class="ref" href="square_1.html#T46" onmouseover="rs('square_1/T46')" onmouseout="rh()">SQUARE_1:46</a></span>;<br/><a NAME="E10:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</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="E11:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E12:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E13:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A97</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E14:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A93</font></span></a>, <a class="ref" href="xreal_1.html#T142" onmouseover="rs('xreal_1/T142')" onmouseout="rh()">XREAL_1:142</a></span>;<br/><a NAME="E15:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</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="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><span class="kw">hence </span><a NAME="E16:51_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></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="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="txt" href="#E8:51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A96</font></span></a>, <a class="txt" href="#E13:51_2_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E75">A98</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A99</font></span>: 
<font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</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="E4:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/><a NAME="E5:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A100</font></span>: 
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A101</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T46" onmouseover="rs('square_1/T46')" onmouseout="rh()">SQUARE_1:46</a></span>;<br/><a NAME="E7:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a></span>;<br/><a NAME="E8:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><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="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><a NAME="E9:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A101</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="E10:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E11:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E12:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">t</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="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></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="E13:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A102</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</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="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E14:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A100</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E15:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>;<br/><a NAME="E16:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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/><span class="kw">hence </span><a NAME="E17:51_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></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="#E1:51_2_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A92</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A98</font></span></a>, <a class="txt" href="#E13:51_2_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A102</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_2_1_1_1_1_1_2_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_2_1_1_1_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:51_2_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</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="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="txt" href="#E1:51_2_1_1_1"><span class="lab"><font color="Green" title="E71">A77</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E4:51_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`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="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_2_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2_1_1_1"><span class="lab"><font color="Green" title="E72">A78</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A103</font></span>: 
(  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font></span><br/><div class="add"><a NAME="E2:51_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A104</font></span>: 
( ( <font color="Maroon" title="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T13" target="_self" onmouseover="rs('jgraph_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><div><span class="lab"><font color="Green" title="E73">A105</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E74">A106</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A107</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A108</font></span>: 
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c7">t</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> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_2_1_1_1_1_1_1"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A107</font></span></a></span>;<br/><a NAME="E4:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A109</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A107</font></span></a>, <a class="txt" href="#E2:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A108</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T14" onmouseover="rs('square_1/T14')" onmouseout="rh()">SQUARE_1:14</a></span>;<br/><a NAME="E6:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</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="E7:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</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="c7">t</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> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E8:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E9:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A110</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E10:51_2_1_1_2_1_1_1_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A107</font></span></a>, <a class="ref" href="xreal_1.html#T139" onmouseover="rs('xreal_1/T139')" onmouseout="rh()">XREAL_1:139</a></span>;<br/><a NAME="E11:51_2_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A111</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E12:51_2_1_1_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A109</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><span class="kw">hence </span><a NAME="E13:51_2_1_1_2_1_1_1_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></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="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="txt" href="#E11:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A111</font></span></a>, <a class="txt" href="#E9:51_2_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A110</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A112</font></span>: 
<font color="Maroon" title="c7">t</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>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E76">A113</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_2_1_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">t</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> 1 implies <font color="Maroon" title="c7">t</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> 1 )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_2_1_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">t</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> 1
 ; <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="c7">t</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> 1</span><br/><a NAME="E2:51_2_1_1_2_1_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A112</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_2_1_1_1_1_1_2_1"/><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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2"><span class="lab"><font color="Green" title="E72">A104</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_2_1_1_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">t</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> 1
 <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:51_2_1_1_2_1_1_1_1_1_2"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E2:51_2_1_1_2"><span class="lab"><font color="Green" title="E72">A104</font></span></a></span>;<br/><a NAME="E4:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A114</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E76">A113</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T14" onmouseover="rs('square_1/T14')" onmouseout="rh()">SQUARE_1:14</a></span>;<br/><a NAME="E6:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</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="E7:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E8:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E9:51_2_1_1_2_1_1_1_1_1_2"/><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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></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="E10:51_2_1_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A115</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><a NAME="E11:51_2_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E77">A114</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_2_1_1_2_1_1_1_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></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="#E1:51_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A106</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E75">A112</font></span></a>, <a class="txt" href="#E10:51_2_1_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E78">A115</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:51_2_1_1_2_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E74">A116</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A117</font></span>: 
<font color="Maroon" title="c7">t</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A118</font></span>: 
( <font color="Maroon" title="c7">t</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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c7">t</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="ref" href="jgraph_2.html#T13" target="_self" onmouseover="rs('jgraph_2/T13')" onmouseout="rh()">Th13</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_2_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">t</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A119</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_2_1_1_2_1_1_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c7">t</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> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2"><span class="lab"><font color="Green" title="E72">A104</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</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="E4:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A120</font></span>: 
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></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="E5:51_2_1_1_2_1_1_2_1_1_1"/>
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E2:51_2_1_1_2"><span class="lab"><font color="Green" title="E72">A104</font></span></a></span>;<br/><a NAME="E6:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A121</font></span>: 
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A120</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E8:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A122</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></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="E9:51_2_1_1_2_1_1_2_1_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A121</font></span></a>, <a class="ref" href="square_1.html#T46" onmouseover="rs('square_1/T46')" onmouseout="rh()">SQUARE_1:46</a></span>;<br/><a NAME="E10:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</font></span></a>, <a class="txt" href="#E3:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A118</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="E11:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E12:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E13:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A123</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E14:51_2_1_1_2_1_1_2_1_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A119</font></span></a>, <a class="ref" href="xreal_1.html#T142" onmouseover="rs('xreal_1/T142')" onmouseout="rh()">XREAL_1:142</a></span>;<br/><a NAME="E15:51_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><span class="kw">hence </span><a NAME="E16:51_2_1_1_2_1_1_2_1_1_1"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></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="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="txt" href="#E8:51_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A122</font></span></a>, <a class="txt" href="#E13:51_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A123</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1_2_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_2_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A124</font></span>: 
<font color="Maroon" title="c7">t</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>  <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"> (  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )</span><br/><div class="add"><a NAME="E2:51_2_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A125</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="txt" href="#E2:51_2_1_1_2"><span class="lab"><font color="Green" title="E72">A104</font></span></a></span>;<br/><a NAME="E3:51_2_1_1_2_1_1_2_1_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E2:51_2"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A124</font></span></a></span>;<br/><a NAME="E4:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A126</font></span>: 
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</font></span></a>, <a class="txt" href="#E2:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A125</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A127</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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 class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T46" onmouseover="rs('square_1/T46')" onmouseout="rh()">SQUARE_1:46</a></span>;<br/><a NAME="E6:51_2_1_1_2_1_1_2_1_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><a NAME="E7:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A127</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="E8:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</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> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E9:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><a NAME="E10:51_2_1_1_2_1_1_2_1_1_2"/><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="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></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="E11:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A128</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E12:51_2_1_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</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">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A126</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/><a NAME="E13:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A117</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>;<br/><a NAME="E14:51_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></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/><span class="kw">hence </span><a NAME="E15:51_2_1_1_2_1_1_2_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></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="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A116</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A124</font></span></a>, <a class="txt" href="#E11:51_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A128</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_2_1_1_2_1_1_2"/>
(  <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> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E4:51_2_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="txt" href="#E1:51_2_1_1_2"><span class="lab"><font color="Green" title="E71">A103</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E5:51_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <font color="Maroon" title="c8">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><span class="kw">hence </span><a NAME="E6:51_2_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E3:51_2_1_1_2"><span class="lab"><font color="Green" title="E73">A105</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:51_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:51"/><span class="lab"><font color="Green" title="E67">A129</font></span>: 
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<a NAME="E8:51"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:51_3"/><span class="lab"><font color="Green" title="E68">A130</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font></span><br/>

<a NAME="E2:51_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A131</font></span>: 
 ex <font color="Olive" title="b1">p8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p8</font> &amp; ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 
;<br/>
<a NAME="E3:51_3_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E2:51_3_1"><span class="lab"><font color="Green" title="E69">A131</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:51_3"/><span class="lab"><font color="Green" title="E69">A132</font></span>: 
 <a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="jgraph_2.html#E18"><span class="lab"><font color="Green" title="E18">Lm1</font></span></a>, <a class="ref" href="revrot_1.html#T19" onmouseover="rs('revrot_1/T19')" onmouseout="rh()">REVROT_1:19</a></span>;<br/>
<a NAME="E3:51_3"/>
( ( <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T5" target="_self" onmouseover="rs('jgraph_2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E4:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A133</font></span>: 
 <a href="euclid.html#K22" title="EUCLID:func.22">1.REAL</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E69">A132</font></span></a></span>;<br/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c6">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">K11</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3"><span class="lab"><font color="Green" title="E70">A133</font></span></a>, <a class="txt" href="#E1:51_3"><span class="lab"><font color="Green" title="E68">A130</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">K01</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T17" target="_self" onmouseover="rs('jgraph_2/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> &amp; <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font> implies <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:51_3"/><span class="lab"><font color="Green" title="E71">A134</font></span>: 
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a>
 <span class="kw">and </span><br/><a NAME="E9:51_3"/><span class="lab"><font color="Green" title="E72">A135</font></span>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a>
 <span class="kw">and </span><br/><a NAME="E10:51_3"/><span class="lab"><font color="Green" title="E73">A136</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/>

<a NAME="E11:51_3"/>
 <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T9" target="_self" onmouseover="rs('jgraph_2/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E12:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A137</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A138</font></span>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E9:51_3"><span class="lab"><font color="Green" title="E72">A135</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c12">p2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3"><span class="lab"><font color="Green" title="E71">A134</font></span></a>, <a class="txt" href="#E9:51_3"><span class="lab"><font color="Green" title="E72">A135</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E15:51_3"/><span class="lab"><font color="Green" title="E76">A139</font></span>: 
<font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">K01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">K11</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">K11</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_3_3"/><span class="lab"><font color="Green" title="E77">A140</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">K11</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">px</font> = <font color="Maroon" title="c13">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:51_3_3"/>
 not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E67">A129</font></span></a>, <a class="txt" href="#E1:51_3_3"><span class="lab"><font color="Green" title="E77">A140</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:51_3_3"/><span class="kw">then </span>
( ( ( ( <font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c14">px</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> <font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Maroon" title="c14">px</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) or ( ( ( <font color="Maroon" title="c14">px</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="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c14">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c14">px</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Maroon" title="c14">px</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xreal_1.html#T25" onmouseover="rs('xreal_1/T25')" onmouseout="rh()">XREAL_1:25</a></span>;<br/>
<a NAME="E5:51_3_3"/><span class="kw">then </span>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K11</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:51_3_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">K11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:51_3"/><span class="lab"><font color="Green" title="E77">A141</font></span>: 
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:51_3"><span class="lab"><font color="Green" title="E71">A134</font></span></a>, <a class="txt" href="#E12:51_3"><span class="lab"><font color="Green" title="E74">A137</font></span></a>, <a class="ref" href="subset_1.html#D4" onmouseover="rs('subset_1/D4')" onmouseout="rh()">SUBSET_1:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> ) or ( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1"/>
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font> or <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3"><span class="lab"><font color="Green" title="E76">A139</font></span></a>, <a class="txt" href="#E16:51_3"><span class="lab"><font color="Green" title="E77">A141</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K01</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/><div class="add"><a NAME="E2:51_3_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A142</font></span>: 
 ex <font color="Olive" title="b1">p7</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="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p7</font> &amp; ( ( <font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p7</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> <font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><a NAME="E3:51_3_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A143</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">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="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> ) or ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1"/>
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  or ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3"><span class="lab"><font color="Green" title="E76">A139</font></span></a>, <a class="txt" href="#E13:51_3"><span class="lab"><font color="Green" title="E75">A138</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_1"/>
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p8</font> &amp; ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><a NAME="E3:51_3_4_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A144</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">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="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_1"><span class="lab"><font color="Green" title="E79">A143</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E4:51_3_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A145</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">qq</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/><span class="lab"><font color="Green" title="E82">A146</font></span>: <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 

<br/>.= 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><div><span class="lab"><font color="Green" title="E83">A147</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E84">A148</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><a NAME="E2:51_3_4_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a>, <a class="txt" href="#E1:51_3_4_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A148</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:51_3_4_1_1_1_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</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="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E8:51_3_4_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A149</font></span>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A144</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E9:51_3_4_1_1_1_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`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="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</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="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E10:51_3_4_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A144</font></span></a>, <a class="txt" href="#E8:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A149</font></span></a>, <a class="txt" href="#E5:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A146</font></span></a>, <a class="txt" href="#E6:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E83">A147</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a>, <a class="ref" href="xcmplx_1.html#T53" onmouseover="rs('xcmplx_1/T53')" onmouseout="rh()">XCMPLX_1:53</a></span>;<br/><a NAME="E11:51_3_4_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E83">A147</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" onmouseover="rs('xcmplx_1/T53')" onmouseout="rh()">XCMPLX_1:53</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_3_4_1_1_1_1_1"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A149</font></span></a>, <a class="txt" href="#E5:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A146</font></span></a>, <a class="txt" href="#E4:51_3_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E81">A145</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2"/><span class="lab"><font color="Green" title="E80">A150</font></span>: 
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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"> contradiction</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E81">A151</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A152</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><a NAME="E2:51_3_4_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a>, <a class="txt" href="#E1:51_3_4_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A152</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E82">A153</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1_2_2_1"/>
( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_2_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E83">A154</font></span>: 
( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_2_1_1"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E83">A154</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_2_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E83">A155</font></span>: 
( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_2_1_2"/><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="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E83">A155</font></span></a>, <a class="ref" href="xreal_1.html#T73" onmouseover="rs('xreal_1/T73')" onmouseout="rh()">XREAL_1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E83">A156</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or ( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1_2_3_1"/>
( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1"><span class="lab"><font color="Green" title="E78">A142</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_3_1_1"/><span class="lab"><font color="Green" title="E84">A157</font></span>: 
( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_3_1_1"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_3_1_1"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_3_1_1"><span class="lab"><font color="Green" title="E84">A157</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_3_1_1"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E84">A158</font></span>: 
( <font color="Maroon" title="c11">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> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_3_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">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>  <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#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a>, <a class="txt" href="#E1:51_3_4_1_1_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A158</font></span></a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_3_1_2"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E4:51_3_4_1_1_1_1_2_3_1_2"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" onmouseover="rs('xcmplx_1/T198')" onmouseout="rh()">XCMPLX_1:198</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_3_1_2"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</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="xcmplx_1.html#T191" onmouseover="rs('xcmplx_1/T191')" onmouseout="rh()">XCMPLX_1:191</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:51_3_4_1_1_1_1_2"/><span class="lab"><font color="Green" title="E84">A159</font></span>: 
 ex <font color="Olive" title="b1">p8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p8</font> &amp; ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A150</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E85">A160</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_1_1_1_2_4"/><span class="lab"><font color="Green" title="E86">A161</font></span>: 
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_3_4_1_1_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A159</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_1_1_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A159</font></span></a>, <a class="txt" href="#E1:51_3_4_1_1_1_1_2_4"><span class="lab"><font color="Green" title="E86">A161</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:51_3_4_1_1_1_1_2"/>
( (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) or  not <font color="Maroon" title="c12">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A150</font></span></a></span>;<br/><a NAME="E8:51_3_4_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A162</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E84">A159</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E9:51_3_4_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_1"><span class="lab"><font color="Green" title="E79">A143</font></span></a>, <a class="ref" href="sppol_2.html#T1" onmouseover="rs('sppol_2/T1')" onmouseout="rh()">SPPOL_2:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A163</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>

;<br/><a NAME="E11:51_3_4_1_1_1_1_2"/>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_1"><span class="lab"><font color="Green" title="E79">A143</font></span></a>, <a class="txt" href="#E8:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E86">A162</font></span></a>, <a class="ref" href="sppol_2.html#T1" onmouseover="rs('sppol_2/T1')" onmouseout="rh()">SPPOL_2:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A164</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>

;<br/><a NAME="E13:51_3_4_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A165</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">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="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="txt" href="#E10:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E87">A163</font></span></a>, <a class="ref" href="xcmplx_1.html#T112" onmouseover="rs('xcmplx_1/T112')" onmouseout="rh()">XCMPLX_1:112</a></span>;<br/><a NAME="E14:51_3_4_1_1_1_1_2"/><span class="lab"><font color="Green" title="E90">A166</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">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="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="txt" href="#E12:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E88">A164</font></span></a>, <a class="txt" href="#E10:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E87">A163</font></span></a>, <a class="ref" href="xcmplx_1.html#T112" onmouseover="rs('xcmplx_1/T112')" onmouseout="rh()">XCMPLX_1:112</a></span>;<br/><a NAME="E15:51_3_4_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A167</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a></span>;<br/><a NAME="E16:51_3_4_1_1_1_1_2"/><span class="lab"><font color="Green" title="E92">A168</font></span>: 
 ex <font color="Olive" title="b1">p9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p9</font> &amp; ( ( <font color="Olive" title="b1">p9</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="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p9</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A150</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E93">A169</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) or ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_7_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1_2_7_1"/>
( ( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A168</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_7_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_7_1_1"/><span class="lab"><font color="Green" title="E94">A170</font></span>: 
( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_7_1_1"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_7_1_1"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E94">A170</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_7_1_1"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_7_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_7_1_2"/><span class="lab"><font color="Green" title="E94">A171</font></span>: 
( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_7_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a>, <a class="txt" href="#E1:51_3_4_1_1_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E94">A171</font></span></a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_7_1_2"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E4:51_3_4_1_1_1_1_2_7_1_2"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" onmouseover="rs('xcmplx_1/T198')" onmouseout="rh()">XCMPLX_1:198</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_7_1_2"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T191" onmouseover="rs('xcmplx_1/T191')" onmouseout="rh()">XCMPLX_1:191</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E18:51_3_4_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">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="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E90">A166</font></span></a></span>;<br/><a NAME="E19:51_3_4_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A172</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E20:51_3_4_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A173</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E91">A167</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><div><span class="lab"><font color="Green" title="E96">A174</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_8_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1_2_8_1"/>
( ( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A168</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_8_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_8_1_1"/><span class="lab"><font color="Green" title="E97">A175</font></span>: 
( <font color="Maroon" title="c12">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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_8_1_1"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_8_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_8_1_1"><span class="lab"><font color="Green" title="E97">A175</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_8_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_8_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_8_1_2"/><span class="lab"><font color="Green" title="E97">A176</font></span>: 
( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c12">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_8_1_2"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_1_1_1_2_8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E4:51_3_4_1_1_1_1_2_8_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2_8_1_2"><span class="lab"><font color="Green" title="E97">A176</font></span></a>, <a class="ref" href="xreal_1.html#T73" onmouseover="rs('xreal_1/T73')" onmouseout="rh()">XREAL_1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_1_1_1_2_8_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; contradiction ) or (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_1_1_1_2_9_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_9_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A177</font></span>: 
( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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>  ) or ( <font color="Maroon" title="c11">p1</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="Maroon" title="c11">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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="txt" href="#E19:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E94">A172</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_1_1_1_2_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_3_4_1_1_1_1_2_9_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></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="#E3:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E82">A153</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_1_1_1_2_9_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E89">A165</font></span></a>, <a class="txt" href="#E21:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E96">A174</font></span></a>, <a class="txt" href="#E2:51_3_4_1_1_1_1_2_9_1_1"><span class="lab"><font color="Green" title="E97">A177</font></span></a>, <a class="ref" href="xreal_1.html#T162" onmouseover="rs('xreal_1/T162')" onmouseout="rh()">XREAL_1:162</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:51_3_4_1_1_1_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="txt" href="#E20:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E95">A173</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><a NAME="E6:51_3_4_1_1_1_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><span class="kw">hence </span><a NAME="E7:51_3_4_1_1_1_1_2_9_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A150</font></span></a>, <a class="txt" href="#E16:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A168</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_1_1_1_2_9_1_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_4_1_1_1_1_2_9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A178</font></span>: 
( ( <font color="Maroon" title="c11">p1</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="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">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">by</span> <span class="lab"><a class="txt" href="#E20:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E95">A173</font></span></a>, <a class="ref" href="xreal_1.html#T143" onmouseover="rs('xreal_1/T143')" onmouseout="rh()">XREAL_1:143</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_1_1_1_2_9_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_1_1_1_2_9_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_3_4_1_1_1_1_2_9_1_2_1"/><span class="kw">then </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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E83">A156</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_1_1_1_2_9_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E89">A165</font></span></a>, <a class="txt" href="#E17:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E93">A169</font></span></a>, <a class="txt" href="#E2:51_3_4_1_1_1_1_2_9_1_2"><span class="lab"><font color="Green" title="E97">A178</font></span></a>, <a class="ref" href="xreal_1.html#T166" onmouseover="rs('xreal_1/T166')" onmouseout="rh()">XREAL_1:166</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E81">A151</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>

;<br/><a NAME="E5:51_3_4_1_1_1_1_2_9_1_2"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E6:51_3_4_1_1_1_1_2_9_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E91">A167</font></span></a>, <a class="txt" href="#E20:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E95">A173</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>;<br/><a NAME="E7:51_3_4_1_1_1_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E85">A160</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E8:51_3_4_1_1_1_1_2_9_1_2"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><span class="kw">hence </span><a NAME="E9:51_3_4_1_1_1_1_2_9_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E80">A150</font></span></a>, <a class="txt" href="#E16:51_3_4_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A168</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:51_3_4_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_3_4_1_1"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/><div class="add"><a NAME="E2:51_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A179</font></span>: 
 ex <font color="Olive" title="b1">p7</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="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p7</font> &amp; ( ( <font color="Olive" title="b1">p7</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="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p7</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p7</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p7</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><a NAME="E3:51_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A180</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_2.html#T14" target="_self" onmouseover="rs('jgraph_2/T14')" onmouseout="rh()">Th14</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> ) or ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1"/>
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  or ( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3"><span class="lab"><font color="Green" title="E76">A139</font></span></a>, <a class="txt" href="#E13:51_3"><span class="lab"><font color="Green" title="E75">A138</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_1"/>
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font></span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p8</font> &amp; ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><a NAME="E3:51_3_4_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A181</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_2"><span class="lab"><font color="Green" title="E79">A180</font></span></a>, <a class="ref" href="jgraph_2.html#T14" target="_self" onmouseover="rs('jgraph_2/T14')" onmouseout="rh()">Th14</a></span>;<br/><a NAME="E4:51_3_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E81">A182</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">qq</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/><span class="lab"><font color="Green" title="E82">A183</font></span>: <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 

<br/>.= 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 

;<br/><div><span class="lab"><font color="Green" title="E83">A184</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E84">A185</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="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"> contradiction</span><br/><a NAME="E2:51_3_4_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a>, <a class="txt" href="#E1:51_3_4_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E84">A185</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:51_3_4_1_2_1_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E8:51_3_4_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A186</font></span>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A181</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E9:51_3_4_1_2_1_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>;<br/><a NAME="E10:51_3_4_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A181</font></span></a>, <a class="txt" href="#E8:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A186</font></span></a>, <a class="txt" href="#E5:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E82">A183</font></span></a>, <a class="txt" href="#E6:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A184</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a>, <a class="ref" href="xcmplx_1.html#T53" onmouseover="rs('xcmplx_1/T53')" onmouseout="rh()">XCMPLX_1:53</a></span>;<br/><a NAME="E11:51_3_4_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A184</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" onmouseover="rs('xcmplx_1/T53')" onmouseout="rh()">XCMPLX_1:53</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_3_4_1_2_1_1_1"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A186</font></span></a>, <a class="txt" href="#E5:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E82">A183</font></span></a>, <a class="txt" href="#E4:51_3_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A182</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E80">A187</font></span>: 
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </span> } </span>  &amp;  not <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( ( ( <font color="Olive" title="b1">p8</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="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p8</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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) </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"> contradiction</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E81">A188</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A189</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="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"> contradiction</span><br/><a NAME="E2:51_3_4_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a>, <a class="txt" href="#E1:51_3_4_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E82">A189</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E82">A190</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1_2_2_1"/>
( ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_2_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E83">A191</font></span>: 
( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_2_1_1"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E83">A191</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_2_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E83">A192</font></span>: 
( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_2_1_2"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E83">A192</font></span></a>, <a class="ref" href="xreal_1.html#T73" onmouseover="rs('xreal_1/T73')" onmouseout="rh()">XREAL_1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E83">A193</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1_2_3_1"/>
( ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2"><span class="lab"><font color="Green" title="E78">A179</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_3_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_3_1_1"/><span class="lab"><font color="Green" title="E84">A194</font></span>: 
( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_3_1_1"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_3_1_1"/><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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_3_1_1"><span class="lab"><font color="Green" title="E84">A194</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_3_1_1"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_3_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_3_1_2"/><span class="lab"><font color="Green" title="E84">A195</font></span>: 
( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c11">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>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_3_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a>, <a class="txt" href="#E1:51_3_4_1_2_1_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A195</font></span></a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_3_1_2"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E4:51_3_4_1_2_1_1_2_3_1_2"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" onmouseover="rs('xcmplx_1/T198')" onmouseout="rh()">XCMPLX_1:198</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_3_1_2"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T191" onmouseover="rs('xcmplx_1/T191')" onmouseout="rh()">XCMPLX_1:191</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E5:51_3_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A196</font></span>: 
 ex <font color="Olive" title="b1">p8</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p8</font> &amp; ( ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p8</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p8</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p8</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A187</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E85">A197</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E86">A198</font></span>: 
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><a NAME="E2:51_3_4_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A196</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_2_1_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A196</font></span></a>, <a class="txt" href="#E1:51_3_4_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E86">A198</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E7:51_3_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E86">A199</font></span>: 
 ex <font color="Olive" title="b1">p9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p9</font> &amp; ( ( <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p9</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <font color="Olive" title="b1">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A187</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E87">A200</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or ( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp;  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_5_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1_2_5_1"/>
( ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A199</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_5_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_5_1_1"/><span class="lab"><font color="Green" title="E88">A201</font></span>: 
( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_5_1_1"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_5_1_1"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_5_1_1"><span class="lab"><font color="Green" title="E88">A201</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_5_1_1"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_5_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_5_1_2"/><span class="lab"><font color="Green" title="E88">A202</font></span>: 
( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_5_1_2"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">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>  <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#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a>, <a class="txt" href="#E1:51_3_4_1_2_1_1_2_5_1_2"><span class="lab"><font color="Green" title="E88">A202</font></span></a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_5_1_2"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><a NAME="E4:51_3_4_1_2_1_1_2_5_1_2"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T198" onmouseover="rs('xcmplx_1/T198')" onmouseout="rh()">XCMPLX_1:198</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_5_1_2"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</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="xcmplx_1.html#T191" onmouseover="rs('xcmplx_1/T191')" onmouseout="rh()">XCMPLX_1:191</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E9:51_3_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E88">A203</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A196</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E10:51_3_4_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <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="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_2"><span class="lab"><font color="Green" title="E79">A180</font></span></a>, <a class="ref" href="sppol_2.html#T1" onmouseover="rs('sppol_2/T1')" onmouseout="rh()">SPPOL_2:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A204</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>

;<br/><a NAME="E12:51_3_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51_3"><span class="lab"><font color="Green" title="E73">A136</font></span></a>, <a class="txt" href="#E3:51_3_4_1_2"><span class="lab"><font color="Green" title="E79">A180</font></span></a>, <a class="txt" href="#E9:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A203</font></span></a>, <a class="ref" href="sppol_2.html#T1" onmouseover="rs('sppol_2/T1')" onmouseout="rh()">SPPOL_2:1</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>

;<br/><a NAME="E14:51_3_4_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A205</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="txt" href="#E11:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A204</font></span></a>, <a class="ref" href="xcmplx_1.html#T112" onmouseover="rs('xcmplx_1/T112')" onmouseout="rh()">XCMPLX_1:112</a></span>;<br/><a NAME="E15:51_3_4_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A206</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E16:51_3_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">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="p3">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A205</font></span></a></span>;<br/><a NAME="E17:51_3_4_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E18:51_3_4_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E19:51_3_4_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A207</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A206</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/><div><span class="lab"><font color="Green" title="E93">A208</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_8_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1_2_8_1"/>
( ( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A199</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_8_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_8_1_1"/><span class="lab"><font color="Green" title="E94">A209</font></span>: 
( <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_8_1_1"/><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="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_8_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_8_1_1"><span class="lab"><font color="Green" title="E94">A209</font></span></a>, <a class="ref" href="xreal_1.html#T72" onmouseover="rs('xreal_1/T72')" onmouseout="rh()">XREAL_1:72</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_8_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_8_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_8_1_2"/><span class="lab"><font color="Green" title="E94">A210</font></span>: 
( <font color="Maroon" title="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_8_1_2"/><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="c12">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> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:51_3_4_1_2_1_1_2_8_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E4:51_3_4_1_2_1_1_2_8_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2_8_1_2"><span class="lab"><font color="Green" title="E94">A210</font></span></a>, <a class="ref" href="xreal_1.html#T73" onmouseover="rs('xreal_1/T73')" onmouseout="rh()">XREAL_1:73</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_4_1_2_1_1_2_8_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; contradiction ) or (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> &amp; contradiction ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_4_1_2_1_1_2_9_1"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9_1_1"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_9_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A211</font></span>: 
( ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">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>  ) or ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A205</font></span></a>, <a class="txt" href="#E15:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A206</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_2_1_1_2_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_3_4_1_2_1_1_2_9_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></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="#E3:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E82">A190</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_2_1_1_2_9_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A205</font></span></a>, <a class="txt" href="#E20:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E93">A208</font></span></a>, <a class="txt" href="#E2:51_3_4_1_2_1_1_2_9_1_1"><span class="lab"><font color="Green" title="E94">A211</font></span></a>, <a class="ref" href="xreal_1.html#T162" onmouseover="rs('xreal_1/T162')" onmouseout="rh()">XREAL_1:162</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:51_3_4_1_2_1_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">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="p2">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="txt" href="#E19:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A207</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" onmouseover="rs('xcmplx_1/T60')" onmouseout="rh()">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><a NAME="E6:51_3_4_1_2_1_1_2_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><span class="kw">hence </span><a NAME="E7:51_3_4_1_2_1_1_2_9_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A187</font></span></a>, <a class="txt" href="#E7:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A199</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9_1_2"><span class="kw">case </span></a><a NAME="E1:51_3_4_1_2_1_1_2_9_1_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_4_1_2_1_1_2_9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A212</font></span>: 
( ( <font color="Maroon" title="c11">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="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">p1</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>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A207</font></span></a>, <a class="ref" href="xreal_1.html#T143" onmouseover="rs('xreal_1/T143')" onmouseout="rh()">XREAL_1:143</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_3_4_1_2_1_1_2_9_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_3_4_1_2_1_1_2_9_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_3_4_1_2_1_1_2_9_1_2_1"/><span class="kw">then </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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A193</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_4_1_2_1_1_2_9_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A205</font></span></a>, <a class="txt" href="#E8:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A200</font></span></a>, <a class="txt" href="#E2:51_3_4_1_2_1_1_2_9_1_2"><span class="lab"><font color="Green" title="E94">A212</font></span></a>, <a class="ref" href="xreal_1.html#T166" onmouseover="rs('xreal_1/T166')" onmouseout="rh()">XREAL_1:166</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A188</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span>

;<br/><a NAME="E5:51_3_4_1_2_1_1_2_9_1_2"/><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="c11">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E6:51_3_4_1_2_1_1_2_9_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">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">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A206</font></span></a>, <a class="txt" href="#E19:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A207</font></span></a>, <a class="ref" href="xcmplx_1.html#T197" onmouseover="rs('xcmplx_1/T197')" onmouseout="rh()">XCMPLX_1:197</a></span>;<br/><a NAME="E7:51_3_4_1_2_1_1_2_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A197</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/><a NAME="E8:51_3_4_1_2_1_1_2_9_1_2"/><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="c12">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><span class="kw">hence </span><a NAME="E9:51_3_4_1_2_1_1_2_9_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A187</font></span></a>, <a class="txt" href="#E7:51_3_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A199</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E22:51_3_4_1_2_1_1_2"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_3_4_1_2"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:51_3"/>
<font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A213</font></span>: 
 <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E10:51"/><span class="lab"><font color="Green" title="E69">A214</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> holds <br/><a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> implies <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_4"/>
<font color="Maroon" title="c7">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/>

<a NAME="E2:51_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A215</font></span>: 
 ex <font color="Olive" title="b1">p4</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">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp; ( (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <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="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or ( <font color="Olive" title="b1">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <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="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) or (  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p4</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p4</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp;  <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="Olive" title="b1">p4</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p4</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> 1 ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<a NAME="E3:51_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A216</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>;<br/>
<a NAME="E4:51_4"/><span class="lab"><font color="Green" title="E72">A217</font></span>: 
 not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E70">A215</font></span></a>, <a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a>, <a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_4_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_1_1"/>
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) or (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_1"/><span class="lab"><font color="Green" title="E73">A218</font></span>: 
( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) or ( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A219</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</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="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4"><span class="lab"><font color="Green" title="E72">A217</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:51_4_1_1_1"/><span class="lab"><font color="Green" title="E75">A220</font></span>: 
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <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> 1 ) or ( 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E70">A215</font></span></a>, <a class="txt" href="#E1:51_4_1_1_1"><span class="lab"><font color="Green" title="E73">A218</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_4_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_1_1_1_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E70">A215</font></span></a>, <a class="txt" href="#E3:51_4_1_1_1"><span class="lab"><font color="Green" title="E75">A220</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_4_1_1_1_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1_1_1"><span class="lab"><font color="Green" title="E74">A219</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_4_1_1_1_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1_1_1"><span class="lab"><font color="Green" title="E74">A219</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:51_4_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_2"/><span class="lab"><font color="Green" title="E73">A221</font></span>: 
(  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><a NAME="E2:51_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A222</font></span>: 
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_4"><span class="lab"><font color="Green" title="E71">A216</font></span></a>, <a class="ref" href="jgraph_2.html#D1" target="_self" onmouseover="rs('jgraph_2/D1')" onmouseout="rh()">Def1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_4_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) or ( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp; <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_1_1_2_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E70">A215</font></span></a>, <a class="txt" href="#E1:51_4_1_1_2"><span class="lab"><font color="Green" title="E73">A221</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_4_1_1_2_1_1_1"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1_1_2"><span class="lab"><font color="Green" title="E74">A222</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:51_4_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_4_1_1_2_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4_1_1_2"><span class="lab"><font color="Green" title="E74">A222</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:51_4_1_1_2"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:51_4"/>
<a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:51"/>
 ex <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">D</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_2.html#K1" title="JGRAPH_2:func.1">Out_In_Sq</a>  &amp; <font color="Olive" title="b1">h</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="jgraph_2.html#T40" target="_self" onmouseover="rs('jgraph_2/T40')" onmouseout="rh()">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:51"/>
 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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds <br/> not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> ) &amp; (  for <font color="Olive" title="b2">r</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  not <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">K0</font> ) &amp; (  for <font color="Olive" title="b2">s</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Kb</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E68">A213</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E69">A214</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
