<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:66"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="jct_misc.html#V1">connected</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="jct_misc.html#D1">JCT_MISC:def 1</a><br/>

<b>thus </b><a NAME="E2:66_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:66"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rcomp_3.html#E1:66_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="borsuk_5.html#T111">BORSUK_5:111</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:66"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="jct_misc.html#V1">connected</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  or ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_2.html#K2">.]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E1:66"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rcomp_3.html#E3:66"><i><font color="Green">E48</font></i></a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1"><b>suppose </b></a><a NAME="E1:66_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  ;<br/><a NAME="E3:66_2_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/><b>hence </b><a NAME="E4:66_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_2"><b>suppose </b></a><a NAME="E1:66_2_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"> <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> </span>)</span> = 
 <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>
<br/>.= 
 <a href="numbers.html#K1">REAL</a> 

;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E1:66_2_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E4:66_2_2"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E5:66_2_2"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_3"><b>suppose </b></a><a NAME="E1:66_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:66_2_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:66_2_3"/>
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:66_2_3"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:66_2_3"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:66_2_3"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_4"><b>suppose </b></a><a NAME="E1:66_2_4"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:66_2_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:66_2_4"/>
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:66_2_4"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:66_2_4"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:66_2_4"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_5"><b>suppose </b></a><a NAME="E1:66_2_5"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:66_2_5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:66_2_5"/>
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:66_2_5"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:66_2_5"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:66_2_5"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_6"><b>suppose </b></a><a NAME="E1:66_2_6"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:66_2_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:66_2_6"/>
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:66_2_6"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:66_2_6"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:66_2_6"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_7"><b>suppose </b></a><a NAME="E1:66_2_7"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_2.html#T8">TOPALG_2:8</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E3:66_2_7"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:66_2_7"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_8"><b>suppose </b></a><a NAME="E1:66_2_8"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_2.html#T10">TOPALG_2:10</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E3:66_2_8"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:66_2_8"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_9"><b>suppose </b></a><a NAME="E1:66_2_9"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_2.html#T11">TOPALG_2:11</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E3:66_2_9"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:66_2_9"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_10"><b>suppose </b></a><a NAME="E1:66_2_10"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="topalg_2.html#V2">convex</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_2.html#T9">TOPALG_2:9</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E3:66_2_10"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:66_2_10"/>
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/></div><b>end;</b></div></div><b>end;</b></div>

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