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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="BORSUK_1:func.17">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#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Olive" title="b2">D</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="BORSUK_1:func.17">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#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Olive" title="b1">D</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="BORSUK_1:func.17">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#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c4">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<font color="Maroon" title="c1">XX</font><span class="kw"> such that </span><a NAME="E2:118"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
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="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) )
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<a NAME="E3:118"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K18" title="BORSUK_1:func.18">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c6">xx</font>, <font color="Maroon" title="c7">xx9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><a NAME="E2:118_1"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
<font color="Maroon" title="c5">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="c5">II</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx9</font>
 <span class="kw">and </span><br/><a NAME="E3:118_1"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:118_1"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx9</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><span class="kw">consider </span><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#K18" title="BORSUK_1:func.18">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E6:118_1"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
<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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">t</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">x9</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">t9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E8:118_1"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
<font color="Maroon" title="c7">xx9</font> <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"><font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c11">t9</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></span>;<br/><a NAME="E9:118_1"/><span class="lab"><font color="Green" title="E79">A7</font></span>: 
( <font color="Maroon" title="c5">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#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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K5" title="BORSUK_1:func.5">]</a></span> &amp; <font color="Maroon" title="c5">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c11">t9</font> <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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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">x9</font></span>)</span>,<font color="Maroon" title="c11">t9</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></span>;<br/><a NAME="E10:118_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t9</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E6:118_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E8:118_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E11:118_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">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="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">xx9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E3:118_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E6:118_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E8:118_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></span>;<br/><span class="kw">hence </span><a NAME="E12:118_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E3:118_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E6:118_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E8:118_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="txt" href="#E9:118_1"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="txt" href="#E4:118_1"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:118"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">THETA</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">II</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T19" target="_self" title="BORSUK_1:th.19">Th19</a></span>;<br/>
<span class="kw">reconsider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<a NAME="E9:118"/>
<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="118_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K7" title="RELSET_1:func.7">.:</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">xt9</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#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></span>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K7" title="RELSET_1:func.7">.:</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">FF</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<span class="kw">consider </span><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#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K18" title="BORSUK_1:func.18">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E4:118_2"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
<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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">W</font>,<font color="Maroon" title="c13">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" target="_self" title="BORSUK_1:th.50">Th50</a></span>;<br/>
<a NAME="E5:118_2"/>
<font color="Maroon" title="c5">II</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">xt9</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_2"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="borsuk_1.html#T24" target="_self" title="BORSUK_1:th.24">Th24</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">GG</font> = <font color="Maroon" title="c11">FF</font> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">"</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K4" title="BORSUK_1:func.4">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E74">A8</font></span></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></span>;<br/>
<a NAME="E7:118_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#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:118_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A10</font></span>: 
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></span>;<br/>
<a NAME="E9:118_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T30" target="_self" title="BORSUK_1:th.30">Th30</a></span>;<br/>
<a NAME="E10:118_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:118_2"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="ref" href="borsuk_1.html#D15" target="_self" title="BORSUK_1:def.15">Def15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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><span class="kw"> such that </span><br/><a NAME="E12:118_2"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">GG</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T67" target="_self" title="BORSUK_1:th.67">Th67</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">H9</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T81" target="_self" title="BORSUK_1:th.81">Th81</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">H99</font> = <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c17">H9</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:118_2"><span class="lab"><font color="Green" title="E75">A9</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c19">H</font> = <font color="Maroon" title="c18">H99</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="E15:118_2"/>
<font color="Maroon" title="c5">II</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">U</font></span>)</span>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T23" target="_self" title="BORSUK_1:th.23">Th23</a></span>;<br/>
<a NAME="E16:118_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> &amp; <font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></span>;<br/>
<a NAME="E17:118_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</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#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c14">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118_2"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:118_2"/>
<font color="Maroon" title="c7">THETA</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:118_2"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">THETA9</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c9">THETA99</font> = <font color="Maroon" title="c8">THETA9</font>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> holds <br/> ( <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K17" title="BORSUK_1:func.17">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#K17" title="BORSUK_1:func.17">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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) )</span><br/>

<span class="kw">let </span><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#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K17" title="BORSUK_1:func.17">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#K17" title="BORSUK_1:func.17">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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">W9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><span class="kw"> such that </span><br/><a NAME="E12:118"/><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T71" target="_self" title="BORSUK_1:th.71">Th71</a></span>;<br/>
<a NAME="E13:118"/>
<font color="Maroon" title="c5">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a>  <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"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></span>;<br/>
<a NAME="E14:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E15:118"/>
<font color="Maroon" title="c4">CH1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E73">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:118"/>
<font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E14:118"><span class="lab"><font color="Green" title="E76">A14</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K17" title="BORSUK_1:func.17">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#K17" title="BORSUK_1:func.17">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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> ) )</span><br/>

<a NAME="E17:118"/><span class="lab"><font color="Green" title="E77">A15</font></span>: 
<font color="Maroon" title="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E73">A1</font></span></a></span>;<br/>
<a NAME="E18:118"/>
<font color="Maroon" title="c5">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a>  <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"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></span>;<br/>
<a NAME="E19:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E20:118"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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="c4">CH1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:118"/>
<font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">1[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:118"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="txt" href="#E17:118"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="borsuk_1.html#T80" target="_self" title="BORSUK_1:th.80">Th80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="BORSUK_1:func.17">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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E22:118"/><span class="lab"><font color="Green" title="E79">A17</font></span>: 
<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#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K18" title="BORSUK_1:func.18">I[01]</a>  holds  <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font></span><br/>

<span class="kw">let </span><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#K18" title="BORSUK_1:func.18">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font></span><br/>

<a NAME="E23:118"/>
<font color="Maroon" title="c5">II</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W9</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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="ref" href="borsuk_1.html#T21" target="_self" title="BORSUK_1:th.21">Th21</a></span>;<br/>
<a NAME="E24:118"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">THETA9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">II</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">THETA9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E25:118"/>
<font color="Maroon" title="c4">CH1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">W9</font>,<font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">W9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:118"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E22:118"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:118"/>
<font color="Maroon" title="c9">THETA99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:118"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E12:118"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E24:118"><span class="lab"><font color="Green" title="E80">A18</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
