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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2: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">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> holds <br/>( <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Olive" title="b1">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> )</span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> implies ( <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Maroon" title="c2">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> ) )</span><br/>



<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff <font color="Maroon" title="c2">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> )</span><br/>

<b>thus </b><a NAME="E2:57"/>
( <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> implies <font color="Maroon" title="c2">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</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">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a> implies <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a></span><br/>

<b>let </b><font color="Maroon" title="c3">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="jct_misc.html#D1" title="JCT_MISC:def.1">JCT_MISC:def 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">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<b>thus </b><a NAME="E2:57_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Olive">b<sub>1</sub></font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:57"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="rcomp_3.html#E1:57_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="borsuk_5.html#T111" title="BORSUK_5:th.111">BORSUK_5:111</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:57"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c2">Y</font> is <a href="jct_misc.html#V1" title="JCT_MISC:attr.1">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2"/>
( <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</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>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E1:57"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="rcomp_3.html#E3:57"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1"><b>suppose </b></a><a NAME="E1:57_2_1"/>
<font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  ;<br/><a NAME="E3:57_2_1"/>
<font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ;<br/><b>hence </b><a NAME="E4:57_2_1"/>
<font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2"><b>suppose </b></a><a NAME="E1:57_2_2"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><a NAME="E2:57_2_2"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E1:57_2_2"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/><a NAME="E4:57_2_2"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E5:57_2_2"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_3"><b>suppose </b></a><a NAME="E1:57_2_3"/>
 ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:57_2_3"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E4:57_2_3"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:57_2_3"><i><font color="Green" title="E47">A5</font></i></a></i>;<br/><a NAME="E6:57_2_3"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:57_2_3"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_4"><b>suppose </b></a><a NAME="E1:57_2_4"/>
 ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:57_2_4"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E4:57_2_4"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:57_2_4"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/><a NAME="E6:57_2_4"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:57_2_4"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_5"><b>suppose </b></a><a NAME="E1:57_2_5"/>
 ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:57_2_5"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E4:57_2_5"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:57_2_5"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/><a NAME="E6:57_2_5"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:57_2_5"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_6"><b>suppose </b></a><a NAME="E1:57_2_6"/>
 ex <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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:57_2_6"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E4:57_2_6"/>
 <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E3:57_2_6"><i><font color="Green" title="E47">A8</font></i></a></i>;<br/><a NAME="E6:57_2_6"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E7:57_2_6"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_7"><b>suppose </b></a><a NAME="E1:57_2_7"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="topalg_2.html#T8" title="TOPALG_2:th.8">TOPALG_2:8</a></i>;<br/><a NAME="E3:57_2_7"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:57_2_7"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_8"><b>suppose </b></a><a NAME="E1:57_2_8"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a>, <a class="ref" href="topalg_2.html#T10" title="TOPALG_2:th.10">TOPALG_2:10</a></i>;<br/><a NAME="E3:57_2_8"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:57_2_8"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_9"><b>suppose </b></a><a NAME="E1:57_2_9"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a>, <a class="ref" href="topalg_2.html#T11" title="TOPALG_2:th.11">TOPALG_2:11</a></i>;<br/><a NAME="E3:57_2_9"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:57_2_9"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_10"><b>suppose </b></a><a NAME="E1:57_2_10"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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 <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</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>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topalg_2.html#V2" title="TOPALG_2:attr.2">convex</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="topalg_2.html#T9" title="TOPALG_2:th.9">TOPALG_2:9</a></i>;<br/><a NAME="E3:57_2_10"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>hence </b><a NAME="E4:57_2_10"/>
<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">X</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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