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<span class="kw">let </span><font color="Maroon" title="c12">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a></span>)</span> &amp; <font color="Maroon" title="c12">p</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#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c12">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="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c12">p</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#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span><br/>

<div><span class="lab"><font color="Green" title="E37">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:62_1"/>
<font color="Maroon" title="c12">p</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="toprealb.html#T21" target="_self" title="TOPREALB:th.21">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:62"/>
<font color="Maroon" title="c12">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="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="pre_topc.html#T25" title="PRE_TOPC:th.25">PRE_TOPC:25</a></span>;<br/>
<span class="kw">then </span>1 <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T12" target="_self" title="TOPREALB:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_3.html#T1" title="JGRAPH_3:th.1">JGRAPH_3:1</a></span>
;<br/>
<a NAME="E6:62"/><span class="kw">then </span>
( <font color="Maroon" title="c12">p</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="c12">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="square_1.html#T41" title="SQUARE_1:th.41">SQUARE_1:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:62"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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