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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>   st <font color="Olive" title="b1">A</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <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> ; <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="c3">A</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:50"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c3">A</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</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="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><div class="add"><a NAME="E2:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c2">b</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="txt" href="#E1:50"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="borsuk_4.html#T32" target="_self" title="BORSUK_4:th.32">Th32</a></span>;<br/><a NAME="E3:50_1_1"/>
 <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">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="borsuk_4.html#T32" target="_self" title="BORSUK_4:th.32">Th32</a></span>;<br/><a NAME="E4:50_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> is   <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  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="treal_1.html#T6" title="TREAL_1:th.6">TREAL_1:6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">BA</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></span>;<br/><a NAME="E6:50_1_1"/>
<font color="Maroon" title="c4">BA</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="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="borsuk_1.html#D14" title="BORSUK_1:def.14">BORSUK_1:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E7:50_1_1"/>
<font color="Maroon" title="c3">A</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="txt" href="#E1:50"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</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="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><div class="add"><a NAME="E2:50_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T29" title="XXREAL_1:th.29">XXREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_1_2"/>
<font color="Maroon" title="c3">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <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></div><span class="kw">end;</span></div>

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