<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">o</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>, <font color="Maroon">p</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">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:101"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 ;<br/>

<b>set </b><font color="Maroon">D</font> =  <a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">cB</font> =  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">Bp</font> =  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">OK</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">D1</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">D2</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>;<br/>
<a NAME="E2:101"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#D6">TOPREALB:def 6</a></i>;<br/>
<a NAME="E3:101"/><i><font color="Green">E74</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/>
<a NAME="E4:101"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:101"/><i><font color="Green">E79</font></i>: 
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T17">TOPREAL9:17</a></i>;<br/>
<a NAME="E6:101"/><i><font color="Green">E80</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/>
<a NAME="E7:101"/><i><font color="Green">E81</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2">Tdisk</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="brouwer.html#T3">BROUWER:3</a></i>;<br/>
<a NAME="E8:101"/><i><font color="Green">E82</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E9:101"/><i><font color="Green">E83</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>set </b><font color="Maroon">TD</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">gg</font> =  <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">xo</font> =  <a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">yo</font> =  <a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">dx</font> =  <a href="jordan.html#K5" target="_self">diffX1_X2_1</a> ;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="jordan.html#K6" target="_self">diffX1_X2_2</a> ;<br/>
<b>set </b><font color="Maroon">fx2</font> =  <a href="jordan.html#K7" target="_self">Proj2_1</a> ;<br/>
<b>set </b><font color="Maroon">fy2</font> =  <a href="jordan.html#K8" target="_self">Proj2_2</a> ;<br/>
<b>reconsider </b><font color="Maroon">rr</font> = <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a>  as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">f1</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">rr</font>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">rr</font> as  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Zf1</font> = <font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zfx2</font> = <a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zfy2</font> = <a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zdx</font> = <a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zdy</font> = <a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zxo</font> = <span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">Zyo</font> = <span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<b>set </b><font color="Maroon">xx</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">yy</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>;<br/>
<a NAME="E12:101"/><i><font color="Green">E109</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E13:101"/><i><font color="Green">E110</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:101"/><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:101"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:101"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_1"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_1"/>
<span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:101"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_2"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_2"/>
<span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:101"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_3"/><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_3"/>
<span class="p1">(<span class="default"><a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:101"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_4"/><i><font color="Green">E178</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_4"/>
<span class="p1">(<span class="default"><a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:101"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_5"/><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:101"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_6"/><i><font color="Green">E183</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:101"/><i><font color="Green">E184</font></i>: 
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span><br/> for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:101_7"/><i><font color="Green">E185</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E2:101_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:101_8"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_8"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:101_8"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:101_8"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:101_8"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <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> <b>by </b><i/>;<br/><a NAME="E9:101_8"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E10:101_8"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E11:101_8"/><b>then </b><i><font color="Green">E193</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:101_8"/><i><font color="Green">E194</font></i>: 
<a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E1"><i><font color="Green">Lemma60</font></i></a></i>;<br/><a NAME="E13:101_8"/><i><font color="Green">E195</font></i>: 
<a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon">r2</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/><a NAME="E14:101_8"/><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E15:101_8"/><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a></i>;<br/><i><font color="Green">E199</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T17" target="_self">Th17</a>, , <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, , <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , , , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/><a NAME="E17:101_8"/><i><font color="Green">E252</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_8_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:101_8_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="complex1.html#T2">COMPLEX1:2</a></i>;<br/><b>hence </b><a NAME="E3:101_8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:101_8"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> as  <a href="pscomp_1.html#V9">continuous</a> <a href="partfun3.html#V1">positive-yielding</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D1">PARTFUN3:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">p1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>;<br/>
<a NAME="E25:101"/><i><font color="Green">E253</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:101"/><i><font color="Green">E254</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:101"/><i><font color="Green">E257</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:101"/><i><font color="Green">E258</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:101_9"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:101_9"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:101_9"/><i><font color="Green">E260</font></i>: 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:101_9"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <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> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">r3</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/><a NAME="E8:101_9"/><i><font color="Green">E262</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="jordan.html#T14" target="_self">Th14</a>, <a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, </i>;<br/><a NAME="E9:101_9"/><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a></i>;<br/><a NAME="E10:101_9"/><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, , </i>;<br/><a NAME="E11:101_9"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E12:101_9"/><b>then </b><i><font color="Green">E265</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:101_9"/><i><font color="Green">E271</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:101_9"/><i><font color="Green">E272</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E273</font></i>: <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, , , <a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/><a NAME="E16:101_9"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan.html#E5:101"><i><font color="Green">E79</font></i></a>, , <a class="ref" href="topreal9.html#T8">TOPREAL9:8</a></i>;<br/><a NAME="E17:101_9"/><b>then </b><i><font color="Green">E274</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jgraph_1.html#T46">JGRAPH_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">o</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8">TOPREAL3:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T8">TOPREAL3:8</a></i>
;<br/><a NAME="E19:101_9"/><b>then </b>
<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">z</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E12:101"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E20:101_9"/>
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , <a class="ref" href="jordan.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> as  <a href="pscomp_1.html#V9">continuous</a> <a href="partfun3.html#V3">nonpositive-yielding</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <b>by </b><i><a class="ref" href="partfun3.html#D3">PARTFUN3:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">pp</font> = <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">x3</font> = <span class="p1">(<span class="default"><a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">y3</font> = <span class="p1">(<span class="default"><a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">X3</font> = <span class="p1">(<span class="default"><a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>, <font color="Maroon">Y3</font> = <span class="p1">(<span class="default"><a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="topreala.html#K2">R2Homeomorphism</a> ;<br/>
<a NAME="E32:101"/><i><font color="Green">E275</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Olive">x</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">y</font> being   <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><b> such that </b><br/><a NAME="E2:101_10"/><i><font color="Green">E276</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="brouwer.html#K3">HC</a> <font color="Maroon">y</font>,<font color="Maroon">p</font>,<font color="Maroon">o</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:101_10"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:101_10"/><i><font color="Green">E278</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E4:101"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r2</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r3</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E5:101_10"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:101_10"/><b>then </b><i><font color="Green">E279</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a>  )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:101_10"/><i><font color="Green">E280</font></i>: 
<a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E2"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<a NAME="E8:101_10"/><i><font color="Green">E281</font></i>: 
<a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E3"><i><font color="Green">Lemma63</font></i></a></i>;<br/>
<a NAME="E9:101_10"/><i><font color="Green">E282</font></i>: 
<a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E1"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E10:101_10"/><i><font color="Green">E283</font></i>: 
<a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K1" target="_self">`1</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:101_10"/><i><font color="Green">E284</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:101_10"/><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="jordan.html#K2" target="_self">`2</a> </span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:101_10"/><i><font color="Green">E286</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">X3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:101_10"/><i><font color="Green">E287</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Y3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:101_10"/><i><font color="Green">E288</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:101_10"/><i><font color="Green">E289</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:101_10"/><i><font color="Green">E290</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:101_10"/><i><font color="Green">E291</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E19:101_10"/><i><font color="Green">E292</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:101_10"/><i><font color="Green">E293</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:101_10"/><i><font color="Green">E294</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:101_10"/><i><font color="Green">E295</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:101_10"/><i><font color="Green">E296</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:101_10"/><i><font color="Green">E297</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<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"><span class="p0">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p5">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p0"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:101_10"/><i><font color="Green">E298</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:101_10"/><i><font color="Green">E299</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:101_10"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan.html#E8:101"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E28:101_10"/><b>then </b><i><font color="Green">E301</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E29:101_10"/><i><font color="Green">E302</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan.html#E16"><i><font color="Green">Lemma89</font></i></a></i>;<br/>
<a NAME="E30:101_10"/><i><font color="Green">E303</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E31:101_10"/><i><font color="Green">E304</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma93</font></i></a></i>;<br/>
<a NAME="E32:101_10"/><i><font color="Green">E305</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, </i>;<br/>
<i><font color="Green">E306</font></i>: <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan.html#T10" target="_self">Th10</a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="jordan.html#T25" target="_self">Th25</a>, , , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E14:101"><i><font color="Green">E115</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<a NAME="E34:101_10"/><i><font color="Green">E307</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E35:101_10"/><i><font color="Green">E308</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E17:101"><i><font color="Green">E120</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E36:101_10"/><i><font color="Green">E309</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E16:101"><i><font color="Green">E118</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E37:101_10"/><i><font color="Green">E310</font></i>: 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T25" target="_self">Th25</a>, , <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, , <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , , <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, , <a class="txt" href="jordan.html#E12:101"><i><font color="Green">E109</font></i></a>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E18:101"><i><font color="Green">E122</font></i></a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<i><font color="Green">E311</font></i>: <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T26" target="_self">Th26</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
<br/>.= 
<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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E12:101"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<a NAME="E39:101_10"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E312</font></i>: <span class="p1">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="partfun3.html#D5">PARTFUN3:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<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"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<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"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</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"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<a NAME="E41:101_10"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E313</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="rfunct_1.html#D4">RFUNCT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K15">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p0">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun3.html#K17">sqrt</a> <span class="p4">(<span class="default"><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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, , , <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan.html#T25" target="_self">Th25</a>, , <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, , <a class="txt" href="jordan.html#E12:101"><i><font color="Green">E109</font></i></a>, , , <a class="ref" href="jordan.html#T30" target="_self">Th30</a>, <a class="txt" href="jordan.html#E15:101"><i><font color="Green">E117</font></i></a>, <a class="ref" href="seq_1.html#D7">SEQ_1:def 7</a></i>
;<br/>
<i><font color="Green">E314</font></i>: <font color="Maroon">X3</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K7" target="_self">Proj2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<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"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="jordan.html#E13:101"><i><font color="Green">E110</font></i></a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<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"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</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"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan.html#T25" target="_self">Th25</a>, , <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="txt" href="jordan.html#E39"><i><font color="Green">Lemma137</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<i><font color="Green">E316</font></i>: <font color="Maroon">Y3</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K8" target="_self">Proj2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<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"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<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"><a href="partfun3.html#K15">-</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"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p5">(<span class="default"><a href="partfun3.html#K17">sqrt</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"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K3" target="_self">diffX2_1</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K5" target="_self">diffX1_X2_1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="partfun3.html#K12">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan.html#K4" target="_self">diffX2_2</a> <font color="Maroon">o</font></span>)</span> <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p5"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p0"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p0">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K13">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="partfun3.html#K14">(#)</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun3.html#K20">/</a> <font color="Maroon">m</font></span>)</span> <a href="partfun3.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="jordan.html#K6" target="_self">diffX1_X2_2</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</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"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</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"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="txt" href="jordan.html#E39"><i><font color="Green">Lemma137</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>
;<br/>
<a NAME="E45:101_10"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</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"><a href="real_1.html#K1">-</a> <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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<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"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</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"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</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"><a href="real_1.html#K1">-</a> <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">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p1">(<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"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><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">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</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"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, , <a class="txt" href="jordan.html#E6:101"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan.html#E7:101"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E5:101"><i><font color="Green">E79</font></i></a>, <a class="ref" href="brouwer.html#T8">BROUWER:8</a></i>
<br/>.= 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Y3</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="jordan.html#T31" target="_self">Th31</a>, <a class="ref" href="jordan.html#T32" target="_self">Th32</a>, <a class="ref" href="topreala.html#D2">TOPREALA:def 2</a></i>
<br/>.= 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">y</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan.html#E13:101"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <span class="p2"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x</font></span><a href="borsuk_1.html#K7">]</a></span>
<b>by </b><i>, <a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E33:101"/>
( <font color="Maroon">X3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">Y3</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">X3</font>,<font color="Maroon">Y3</font></span><a href="funct_3.html#K14">:&gt;</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow12.html#T41">YELLOW12:41</a></i>;<br/>
<a NAME="E35:101"/>
for <font color="Olive">pp</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>  st <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">pp</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> st <br/>( <font color="Olive">pp</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">pp</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:101_11"/><i><font color="Green">E317</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:101_11"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">pp</font>, <font color="Maroon">fp</font> = <font color="Maroon">p</font> as   <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> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E4:101_11"/><i><font color="Green">E319</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">pp</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E4:101"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:101_11"/><i><font color="Green">E320</font></i>: 
<font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:101_11"/><i><font color="Green">E321</font></i>: 
<font color="Maroon">V1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E8:101_11"/><i><font color="Green">E322</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">pp</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a></i>;<br/>
<a NAME="E9:101_11"/>
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#T70">TOPS_2:70</a>, <a class="ref" href="topreala.html#T56">TOPREALA:56</a></i>;<br/>
<a NAME="E10:101_11"/><b>then </b><i><font color="Green">E323</font></i>: 
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T16" target="_self">Th16</a>, <a class="ref" href="topgrp_1.html#T25">TOPGRP_1:25</a></i>;<br/>
<a NAME="E11:101_11"/><i><font color="Green">E324</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:101_11"/><i><font color="Green">E325</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:101_11"/><b>then </b><i><font color="Green">E326</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> 
 
;<br/>
<a NAME="E14:101_11"/><b>then </b><i><font color="Green">E327</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E15:101_11"/><i><font color="Green">E328</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="topreala.html#K2">R2Homeomorphism</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a>, <a class="ref" href="topreala.html#T56">TOPREALA:56</a></i>;<br/>
<i><font color="Green">E329</font></i>: <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
;<br/>
<a NAME="E17:101_11"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E18:101_11"/><b>then </b><i><font color="Green">E330</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E12:101"><i><font color="Green">E109</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E19:101_11"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:101_11_2"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:101_11_2"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan.html#T20" target="_self">Th20</a>, <a class="txt" href="jordan.html#E14:101"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:101_11"/><b>then </b><i><font color="Green">E333</font></i>: 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="topreala.html#K2">R2Homeomorphism</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T23" target="_self">Th23</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E21:101_11"/>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E23"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:101_11"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E23:101_11"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13:101"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E25:101_11"/><i><font color="Green">E335</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E26:101_11"/><i><font color="Green">E336</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E27:101_11"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font>
 <b>by </b><i>, , <a class="ref" href="jordan.html#T22" target="_self">Th22</a>, , <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon">WW</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E29:101_11"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">WW</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E30:101_11"/><i><font color="Green">E342</font></i>: 
<font color="Maroon">WW</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>consider </b><font color="Maroon">SF</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E32:101_11"/><i><font color="Green">E343</font></i>: 
<font color="Maroon">WW</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">SF</font>
 <b>and </b><br/><a NAME="E33:101_11"/><i><font color="Green">E344</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> holds <br/> ex <font color="Olive">X1</font>, <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="jordan.html#E16:101"><i><font color="Green">E118</font></i></a>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<a NAME="E34:101_11"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E14:101"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E36:101_11"/><i><font color="Green">E345</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fp</font>,<font color="Maroon">p1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E37:101_11"/><i><font color="Green">E348</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font>
 <b>by </b><i><a class="txt" href="jordan.html#E17:101"><i><font color="Green">E120</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E39:101_11"/><i><font color="Green">E349</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E40:101_11"/>
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E41:101_11"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">ZZ</font> = <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T25" target="_self">Th25</a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>take </b>
<font color="Maroon">XX</font>
;<br/>

<a NAME="E43:101_11"/>
<font color="Maroon">pp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E18:101"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E44:101_11"/>
<font color="Maroon">pp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E45:101_11"/>
<font color="Maroon">XX</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E46:101_11"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">XX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E48:101_11"/><i><font color="Green">E353</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 <b>and </b><br/><a NAME="E49:101_11"/><i><font color="Green">E354</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">fa</font> = <font color="Maroon">fp</font> as   <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> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<a NAME="E51:101_11"/><i><font color="Green">E355</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E52:101_11"/><i><font color="Green">E356</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">a</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="jordan.html#E9:101"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan.html#E4:101"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E53:101_11"/>
<font color="Maroon">fa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E18:101"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E54:101_11"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E55:101_11"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E56:101_11"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E57:101_11"/><i><font color="Green">E361</font></i>: 
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E58:101_11"/><i><font color="Green">E362</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, <a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>;<br/>
<a NAME="E59:101_11"/><i><font color="Green">E363</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7">Tcircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E60:101_11"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">WW</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E17:101"><i><font color="Green">E120</font></i></a>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E61:101_11"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">WW</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p4">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E62:101_11"/><b>then </b>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><span class="p4"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p5">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p5">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">o</font>,<font color="Maroon">r</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E63:101_11"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E64:101_11"/><b>then </b>
<a href="topreala.html#K2">R2Homeomorphism</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a1</font></span><a href="borsuk_1.html#K7">]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E15:101"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E65:101_11"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <a href="topreala.html#K2">R2Homeomorphism</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E66:101_11"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreala.html#K2">R2Homeomorphism</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">fa</font>,<font color="Maroon">a1</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T21" target="_self">Th21</a>, , <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, <a class="ref" href="partfun3.html#T1">PARTFUN3:1</a></i>;<br/>
<a NAME="E67:101_11"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E22"><i><font color="Green">Lemma99</font></i></a></i>;<br/>
<b>hence </b><a NAME="E68:101_11"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="jordan.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan.html#T26" target="_self">Th26</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E36:101"/>
 <a href="jordan.html#K10" target="_self">RotateCircle</a> <font color="Maroon">o</font>,<font color="Maroon">r</font>,<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


</div>
