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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^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">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/> <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">A9</font> = <font color="Maroon" title="c1">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> = <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:67"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/>

<a NAME="E4:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c3">A9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T40" target="_self" title="BORSUK_5:th.40">Th63</a></span>;<br/>
<a NAME="E5:67"/>
<font color="Maroon" title="c4">C</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_5.html#T42" target="_self" title="BORSUK_5:th.42">Th66</a></span>;<br/>
<a NAME="E6:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="tops_1.html#T5" title="TOPS_1:th.5">TOPS_1:5</a>, <a class="ref" href="xxreal_1.html#T45" title="XXREAL_1:th.45">XXREAL_1:45</a></span>;<br/>
<a NAME="E7:67"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="borsuk_5.html#T43" target="_self" title="BORSUK_5:th.43">Th67</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_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="E1:67_1"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A9</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="borsuk_5.html#T1" target="_self" title="BORSUK_5:th.1">Th2</a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="borsuk_5.html#T34" target="_self" title="BORSUK_5:th.34">Th57</a>, <a class="ref" href="borsuk_5.html#T45" target="_self" title="BORSUK_5:th.45">Th71</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></div><span class="kw">end;</span></div>

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