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<b>let </b><font color="Maroon" title="c1">E</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) implies  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">E</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  )</span><br/>

<b>given </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><a NAME="E2:44"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">E</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c4">T</font> = <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">E</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="tsep_1.html#T7" title="TSEP_1:th.7">TSEP_1:7</a></i>;<br/>
<a NAME="E4:44"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">S</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E2:44"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T6" title="TREAL_1:th.6">TREAL_1:6</a></i>;<br/>
<a NAME="E6:44"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:44"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E7:44"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a>, <a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>;<br/>
<a NAME="E8:44"/>
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="borsuk_4.html#E2:44"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="treal_1.html#T20" title="TREAL_1:th.20">TREAL_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:44"/>
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">E</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 <b>by </b><i><a class="txt" href="borsuk_4.html#E7:44"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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