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<b>let </b><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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green" title="E36">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:62"/>
<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>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:62"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>then </b>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> 
<b>by </b><i><a class="ref" href="toprealb.html#T12" target="_self" title="TOPREALB:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></i>
;<br/>
<a NAME="E5:62"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`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#K21" title="EUCLID:func.21">`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 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:62"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon" title="c12">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:62_2"/><b>then </b>
<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> 
 <b>by </b><i><a class="txt" href="toprealb.html#E2:62"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E3:62_2"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E1:62"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="toprealb.html#T22" target="_self" title="TOPREALB:th.22">Th22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><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> 
 <b>by </b><i><a class="txt" href="toprealb.html#E2:62"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="toprealb.html#E5:62"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>


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