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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Olive" title="b1">a</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Olive" title="b3">c</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Olive" title="b3">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c2">a</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )</span><br/>



<b>assume </b><a NAME="E1:94"/><i><font color="Green" title="E61">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c2">a</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="seq_4.html#K1" title="SEQ_4: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="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#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">B</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c2">a</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>, <font color="Maroon" title="c6">C</font> = <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>, <font color="Maroon" title="c7">D</font> =  <a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c8">E</font> = <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</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>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:94"/><i><font color="Green" title="E62">A2</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></i>;<br/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c1">A</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">E</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="borsuk_5.html#T61" target="_self" title="BORSUK_5:th.61">Th61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T50" title="PRE_TOPC:th.50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T97" target="_self" title="BORSUK_5:th.97">Th97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c3">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T55" target="_self" title="BORSUK_5:th.55">Th55</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T29" target="_self" title="BORSUK_5:th.29">Th29</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
<b>by </b><i><a class="txt" href="#E3:94"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/>
<b>hence </b><a NAME="E5:94"/>
 <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#K4" title="RCOMP_1:func.4">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c4">c</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 ; <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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