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<b>let </b><font color="Maroon" title="c1">XX</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">XX</font><br/> for <font color="Olive" title="b2">D</font> being  <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b1">X</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a> <font color="Maroon" title="c1">XX</font> holds <br/> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Olive" title="b2">D</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a>  <a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Olive" title="b2">D</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">XX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Maroon" title="c2">X</font>  st <font color="Maroon" title="c2">X</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a> <font color="Maroon" title="c1">XX</font> holds <br/> <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Olive" title="b1">D</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a>  <a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Olive" title="b1">D</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">D</font> be   <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a> <font color="Maroon" title="c1">XX</font> implies  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a>  <a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/>





<a NAME="E1:107"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">XX</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">II</font> = <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span>)</span></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/>
<b>given </b><font color="Maroon" title="c5">CH1</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<font color="Maroon" title="c1">XX</font><b> such that </b><a NAME="E4:107"/><i><font color="Green" title="E76">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font> holds <br/> ( <font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> &amp; ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b2">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D21" target="_self" title="BORSUK_1:def.21">BORSUK_1:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font> <a href="borsuk_1.html#R2" title="BORSUK_1:pred.2">is_an_SDR_of</a>  <a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">xx</font>, <font color="Maroon" title="c7">xx'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><a NAME="E2:107_1"/><i><font color="Green" title="E77">A2</font></i>: 
<font color="Maroon" title="c4">II</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">II</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx'</font>
 <b>and </b><br/><a NAME="E3:107_1"/><i><font color="Green" title="E78">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c8">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font>, <font color="Maroon" title="c9">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> <b> such that </b><br/><a NAME="E5:107_1"/><i><font color="Green" title="E79">A4</font></i>: 
<font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">x'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font>, <font color="Maroon" title="c11">t'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> <b> such that </b><br/><a NAME="E7:107_1"/><i><font color="Green" title="E80">A5</font></i>: 
<font color="Maroon" title="c7">xx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">x'</font>,<font color="Maroon" title="c11">t'</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></i>;<br/><a NAME="E8:107_1"/><i><font color="Green" title="E81">A6</font></i>: 
( <font color="Maroon" title="c4">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c9">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c4">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x'</font>,<font color="Maroon" title="c11">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x'</font></span>)</span>,<font color="Maroon" title="c11">t'</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></i>;<br/><a NAME="E9:107_1"/><b>then </b>
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t'</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x'</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:107_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E5:107_1"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E7:107_1"><i><font color="Green" title="E80">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E10:107_1"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:107_1"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E5:107_1"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E7:107_1"><i><font color="Green" title="E80">A5</font></i></a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></i>;<br/><a NAME="E11:107_1"/><b>then </b><i><font color="Green" title="E82">A7</font></i>: 
( <font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x'</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:107"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E5:107_1"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E7:107_1"><i><font color="Green" title="E80">A5</font></i></a></i>;<br/><a NAME="E12:107_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E13:107_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:107_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E3:107_1"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E5:107_1"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E7:107_1"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="borsuk_1.html#E8:107_1"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="borsuk_1.html#E11:107_1"><i><font color="Green" title="E82">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">THETA</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:107"/><i><font color="Green" title="E77">A8</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">THETA</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self" title="BORSUK_1:th.19">Th19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">THETA</font> = <font color="Maroon" title="c6">THETA</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<a NAME="E9:107"/>
<font color="Maroon" title="c7">THETA</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">WT</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3" target="_self" title="BORSUK_1:def.3">BORSUK_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">THETA</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">WT</font>  ex <font color="Olive" title="b2">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">WT</font> st <font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">xt'</font> = <font color="Maroon" title="c8">WT</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c10">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">THETA</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">WT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">WT</font> st <font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">G</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">FF</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">W</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>, <font color="Maroon" title="c13">T</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> <b> such that </b><br/><a NAME="E4:107_2"/><i><font color="Green" title="E78">A9</font></i>: 
<font color="Maroon" title="c8">WT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c12">W</font>,<font color="Maroon" title="c13">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></i>;<br/>
<a NAME="E5:107_2"/>
<font color="Maroon" title="c4">II</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">xt'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">T</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:107_2"><i><font color="Green" title="E78">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T24" target="_self" title="BORSUK_1:th.24">Th24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">GG</font> = <font color="Maroon" title="c11">FF</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">"</a> <font color="Maroon" title="c10">G</font> as    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">T</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_1.html#E7:107"><i><font color="Green" title="E77">A8</font></i></a>, <a class="ref" href="borsuk_1.html#T20" target="_self" title="BORSUK_1:th.20">Th20</a>, <a class="ref" href="borsuk_1.html#T39" target="_self" title="BORSUK_1:th.39">Th39</a></i>;<br/>
<a NAME="E7:107_2"/>
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:107_2"/><b>then </b><i><font color="Green" title="E79">A10</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<a NAME="E9:107_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self" title="BORSUK_1:th.30">Th30</a></i>;<br/>
<a NAME="E10:107_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E8:107_2"><i><font color="Green" title="E79">A10</font></i></a>, <a class="ref" href="borsuk_1.html#D15" target="_self" title="BORSUK_1:def.15">Def15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">U</font> being    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c16">V</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c13">T</font><b> such that </b><br/><a NAME="E12:107_2"/><i><font color="Green" title="E80">A11</font></i>: 
<span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">GG</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T67" target="_self" title="BORSUK_1:th.67">Th67</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">H'</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">U</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c12">W</font> <b>by </b><i><a class="ref" href="borsuk_1.html#T81" target="_self" title="BORSUK_1:th.81">Th81</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">H''</font> = <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c17">H'</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">WT</font> <b>by </b><i><a class="txt" href="borsuk_1.html#E4:107_2"><i><font color="Green" title="E78">A9</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c19">H</font> = <font color="Maroon" title="c18">H''</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">G</font></span><br/>

<a NAME="E15:107_2"/><i><font color="Green" title="E81">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">GG</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></i>;<br/>
<a NAME="E16:107_2"/>
<font color="Maroon" title="c4">II</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">U</font></span>)</span>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self" title="BORSUK_1:th.23">Th23</a></i>;<br/>
<a NAME="E17:107_2"/><b>then </b><i><font color="Green" title="E82">A13</font></i>: 
<font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E7:107"><i><font color="Green" title="E77">A8</font></i></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>;<br/>
<a NAME="E18:107_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">CH1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">GG</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E12:107_2"><i><font color="Green" title="E80">A11</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E19:107_2"/>
<font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E15:107_2"><i><font color="Green" title="E81">A12</font></i></a>, <a class="txt" href="borsuk_1.html#E17:107_2"><i><font color="Green" title="E82">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">THETA'</font> = <font color="Maroon" title="c7">THETA</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<b>take </b><font color="Maroon" title="c9">THETA''</font> = <font color="Maroon" title="c8">THETA'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D21" target="_self" title="BORSUK_1:def.21">BORSUK_1:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> holds <br/> ( <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> &amp; ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies for <font color="Olive" title="b2">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c10">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> &amp; <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> &amp; ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies for <font color="Olive" title="b1">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c11">W'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E12:107"/><i><font color="Green" title="E78">A14</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">W'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T71" target="_self" title="BORSUK_1:th.71">Th71</a></i>;<br/>
<a NAME="E13:107"/>
( <font color="Maroon" title="c5">CH1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W'</font> &amp; <font color="Maroon" title="c4">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:107"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></i>;<br/>
<a NAME="E14:107"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E7:107"><i><font color="Green" title="E77">A8</font></i></a>, <a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E15:107"/>
<font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> &amp; ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies for <font color="Olive" title="b1">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> ) )</span><br/>

<a NAME="E16:107"/>
<font color="Maroon" title="c4">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></i>;<br/>
<a NAME="E17:107"/><b>then </b><i><font color="Green" title="E79">A15</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E7:107"><i><font color="Green" title="E77">A8</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E18:107"/>
<font color="Maroon" title="c5">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:107"><i><font color="Green" title="E76">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:107"/>
<font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a> </span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E17:107"><i><font color="Green" title="E79">A15</font></i></a>, <a class="ref" href="borsuk_1.html#T80" target="_self" title="BORSUK_1:th.80">Th80</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies for <font color="Olive" title="b1">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> )</span><br/>

<b>assume </b><a NAME="E20:107"/><i><font color="Green" title="E80">A16</font></i>: 
<font color="Maroon" title="c10">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font></span><br/>

<b>let </b><font color="Maroon" title="c12">T</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font></span><br/>

<a NAME="E21:107"/>
<font color="Maroon" title="c11">W'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="txt" href="borsuk_1.html#E20:107"><i><font color="Green" title="E80">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></i>;<br/>
<a NAME="E22:107"/><b>then </b>
( <font color="Maroon" title="c5">CH1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W'</font>,<font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W'</font> &amp; <font color="Maroon" title="c4">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W'</font>,<font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:107"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></i>;<br/>
<a NAME="E23:107"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c11">W'</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E7:107"><i><font color="Green" title="E77">A8</font></i></a>, <a class="txt" href="borsuk_1.html#E12:107"><i><font color="Green" title="E78">A14</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E24:107"/>
<font color="Maroon" title="c9">THETA''</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
