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<b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">cn</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:133"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="jgraph_4.html#K8" target="_self">FanS</a> <font color="Maroon">cn</font>,<font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:133_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">cn</font> &amp; <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><b>then </b> <a href="jgraph_4.html#K8" target="_self">FanS</a> <font color="Maroon">cn</font>,<font color="Maroon">q</font> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p5"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</a></i>
;<br/><b>hence </b><a NAME="E3:133_1"/>
<span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p0"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">cn</font></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:133"/>
<font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<b>hence </b><a NAME="E4:133"/>
<span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>


</div>
