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<b>let </b><font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c19">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c18">p</font>, <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c19">q</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> </span><br/>

<b>set </b><font color="Maroon" title="c20">P</font> =  <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c18">p</font>;<br/>
<b>set </b><font color="Maroon" title="c21">Q</font> =  <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c19">q</font>;<br/>
<b>set </b><font color="Maroon" title="c22">D</font> =  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">p2</font> = <font color="Maroon" title="c18">p</font>, <font color="Maroon" title="c24">q2</font> = <font color="Maroon" title="c19">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<a NAME="E2:69"/>
( <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font> &amp; <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">A</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c26">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="toprealb.html#T9" target="_self" title="TOPREALB:th.9">Th9</a></i>;<br/>
<i><font color="Green" title="E42">A1</font></i>:  <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c18">p</font> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,1</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">A</font>
<b>by </b><i><a class="txt" href="toprealb.html#E34"><i><font color="Green" title="E23">Lm14</font></i></a>, <a class="ref" href="toprealb.html#T23" target="_self" title="TOPREALB:th.23">Th23</a>, <a class="ref" href="euclid.html#T58" title="EUCLID:th.58">EUCLID:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c23">p2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>
;<br/>
 <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c19">q</font> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,1</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">B</font>
<b>by </b><i><a class="txt" href="toprealb.html#E34"><i><font color="Green" title="E23">Lm14</font></i></a>, <a class="ref" href="toprealb.html#T23" target="_self" title="TOPREALB:th.23">Th23</a>, <a class="ref" href="euclid.html#T58" title="EUCLID:th.58">EUCLID:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>,<font color="Maroon" title="c5">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">q2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>
;<br/>
<b>hence </b><a NAME="E6:69"/>
 <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c18">p</font>, <a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <font color="Maroon" title="c19">q</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E4:69"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="toprealb.html#E34"><i><font color="Green" title="E23">Lm14</font></i></a>, <a class="ref" href="borsuk_4.html#T78" title="BORSUK_4:th.78">BORSUK_4:78</a>, <a class="ref" href="euclid.html#T58" title="EUCLID:th.58">EUCLID:58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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