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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font> be   <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> ; <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="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> implies <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )</span><br/>

<a NAME="E1:51"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">S</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">S</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:51"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><br/>

<a NAME="E4:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> is  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T23" title="TREAL_1:th.23">TREAL_1:23</a></span>;<br/>
<a NAME="E5:51"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T48" target="_self" title="BORSUK_4:th.48">Th48</a></span>;<br/>
<a NAME="E6:51"/>
 <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="c1">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span>
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="topmetr.html#T31" title="TOPMETR:th.31">TOPMETR:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="compts_1.html#T17" title="COMPTS_1:th.17">COMPTS_1:17</a>, <a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</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/>


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