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<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="firebrick">::  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#K16" title="BORSUK_1:func.16">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#K11" title="BORSUK_1:func.11">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="firebrick">::  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#K16" title="BORSUK_1:func.16">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#K11" title="BORSUK_1:func.11">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="firebrick">::  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#K16" title="BORSUK_1:func.16">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#K11" title="BORSUK_1:func.11">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">XX</font><span class="kw"> such that </span><a NAME="E2:76"/><span class="lab"><font color="Green" title="E49">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[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> <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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> ) )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_1.html#D18" target="_self" title="BORSUK_1:def.18">BORSUK_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="borsuk_1.html#K16" title="BORSUK_1:func.16">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#K11" title="BORSUK_1:func.11">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<a NAME="E3:76"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</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#K17" title="BORSUK_1:func.17">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#D2" target="_self" title="BORSUK_1:def.2">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K17" title="BORSUK_1:func.17">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" target="_self" title="BORSUK_1:def.2">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span><span class="kw"> such that </span><a NAME="E2:76_1"/><span class="lab"><font color="Green" title="E50">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="E3:76_1"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:76_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#T15" title="FUNCT_2:th.15">FUNCT_2:15</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#K17" title="BORSUK_1:func.17">I[01]</a><span class="kw"> such that </span><br/><a NAME="E6:76_1"/><span class="lab"><font color="Green" title="E53">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">t</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#T10" target="_self" title="BORSUK_1:th.10">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#K17" title="BORSUK_1:func.17">I[01]</a><span class="kw"> such that </span><br/><a NAME="E8:76_1"/><span class="lab"><font color="Green" title="E54">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c11">t9</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#T10" target="_self" title="BORSUK_1:th.10">Th50</a></span>;<br/><a NAME="E9:76_1"/><span class="lab"><font color="Green" title="E55">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#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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K4" title="BORSUK_1:func.4">]</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#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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K4" title="BORSUK_1:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T58" title="EQREL_1:th.58">EQREL_1:58</a></span>;<br/><a NAME="E10:76_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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="#E2:76_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E8:76_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E11:76_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="#E2:76"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:76_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E8:76_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="borsuk_1.html#T35" target="_self" title="BORSUK_1:th.35">Th78</a></span>;<br/><span class="kw">hence </span><a NAME="E12:76_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E3:76_1"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E6:76_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E8:76_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E9:76_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</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#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:76"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="eqrel_1.html#T56" title="EQREL_1:th.56">EQREL_1:56</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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> ;<br/>
<a NAME="E9:76"/>
<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="76_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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="borsuk_1.html#D1" target="_self" title="BORSUK_1:def.1">BORSUK_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K17" title="BORSUK_1:func.17">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#D2" target="_self" title="BORSUK_1:def.2">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="firebrick">::  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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">XX</font>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K17" title="BORSUK_1:func.17">I[01]</a><span class="kw"> such that </span><br/><a NAME="E4:76_2"/><span class="lab"><font color="Green" title="E51">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c12">W</font>,<font color="Maroon" title="c13">T</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#T10" target="_self" title="BORSUK_1:th.10">Th50</a></span>;<br/>
<a NAME="E5:76_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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_2"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="eqrel_1.html#T60" title="EQREL_1:th.60">EQREL_1:60</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#K1" title="BORSUK_1:func.1">"</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="borsuk_1.html#T4" target="_self" title="BORSUK_1:th.4">Th39</a>, <a class="ref" href="eqrel_1.html#T57" title="EQREL_1:th.57">EQREL_1:57</a></span>;<br/>
<a NAME="E7:76_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#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:76_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">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#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D7" target="_self" title="BORSUK_1:def.7">Def10</a></span>;<br/>
<a NAME="E9:76_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="eqrel_1.html#T66" title="EQREL_1:th.66">EQREL_1:66</a></span>;<br/>
<a NAME="E10:76_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="#E8:76_2"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="borsuk_1.html#D12" target="_self" title="BORSUK_1:def.12">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="E12:76_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</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#T25" target="_self" title="BORSUK_1:th.25">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#T38" target="_self" title="BORSUK_1:th.38">Th81</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">H99</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">H9</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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="#E4:76_2"><span class="lab"><font color="Green" title="E51">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="firebrick">::  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="E15:76_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#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</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> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p4">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T59" title="EQREL_1:th.59">EQREL_1:59</a></span>;<br/>
<a NAME="E16:76_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="funct_1.html#T75" title="FUNCT_1:th.75">FUNCT_1:75</a>, <a class="ref" href="relat_1.html#T126" title="RELAT_1:th.126">RELAT_1:126</a></span>;<br/>
<a NAME="E17:76_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c15">U</font>,<font color="Maroon" title="c16">V</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</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#K12" title="BORSUK_1:func.12">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="#E12:76_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:76_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="#E16:76_2"><span class="lab"><font color="Green" title="E54">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="firebrick">::  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#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="firebrick">:: according to </font></span><a class="ref" href="borsuk_1.html#D18" target="_self" title="BORSUK_1:def.18">BORSUK_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[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> <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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K16" title="BORSUK_1:func.16">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#K16" title="BORSUK_1:func.16">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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K11" title="BORSUK_1:func.11">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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="firebrick">::  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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[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> <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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K16" title="BORSUK_1:func.16">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#K16" title="BORSUK_1:func.16">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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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="E12:76"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#T29" target="_self" title="BORSUK_1:th.29">Th71</a></span>;<br/>
<a NAME="E13:76"/>
<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#K18" title="BORSUK_1:func.18">0[01]</a>) <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"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a></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="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T58" title="EQREL_1:th.58">EQREL_1:58</a></span>;<br/>
<a NAME="E14:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[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> <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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[01]</a></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="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E15:76"/>
<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#K18" title="BORSUK_1:func.18">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="#E2:76"><span class="lab"><font color="Green" title="E49">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:76"/>
<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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K18" title="BORSUK_1:func.18">0[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> <font color="Maroon" title="c10">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E14:76"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K16" title="BORSUK_1:func.16">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#K16" title="BORSUK_1:func.16">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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font> ) )</span><br/>

<a NAME="E17:76"/><span class="lab"><font color="Green" title="E53">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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="#E2:76"><span class="lab"><font color="Green" title="E49">A1</font></span></a></span>;<br/>
<a NAME="E18:76"/>
<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">1[01]</a>) <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"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></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="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T58" title="EQREL_1:th.58">EQREL_1:58</a></span>;<br/>
<a NAME="E19:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[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> <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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></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="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E20:76"/>
<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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[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">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" title="BORSUK_1:func.15">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:76"/>
<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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">1[01]</a></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K16" title="BORSUK_1:func.16">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:76"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E17:76"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="ref" href="borsuk_1.html#T37" target="_self" title="BORSUK_1:th.37">Th80</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K16" title="BORSUK_1:func.16">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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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="E22:76"/><span class="lab"><font color="Green" title="E55">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#K16" title="BORSUK_1:func.16">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K17" title="BORSUK_1:func.17">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Olive" title="b1">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K17" title="BORSUK_1:func.17">I[01]</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">W</font></span><br/>

<a NAME="E23:76"/>
<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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</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="txt" href="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="ref" href="eqrel_1.html#T58" title="EQREL_1:th.58">EQREL_1:58</a></span>;<br/>
<a NAME="E24:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c11">W9</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</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="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E25:76"/>
<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="#E2:76"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E22:76"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="ref" href="borsuk_1.html#T36" target="_self" title="BORSUK_1:th.36">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:76"/>
<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#K4" title="BORSUK_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">W</font>,<font color="Maroon" title="c12">T</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">]</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="#E7:76"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E12:76"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E24:76"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
