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

<span class="kw">let </span><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#K2" title="TOPALG_2:func.2">R^1</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">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="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>  )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:63"/>
( <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="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</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="c2">Y</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">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="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> </span><br/>

<span class="kw">let </span><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> ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:63_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="borsuk_5.html#T111" title="BORSUK_5:th.111">BORSUK_5:111</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>

<span class="kw">assume </span><a NAME="E3:63"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</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="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="63_2"><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="E1:63_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#NR3" title="XXREAL_0:NR.3">&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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</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#NR3" title="XXREAL_0:NR.3">&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#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> ) 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#NR3" title="XXREAL_0:NR.3">&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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  ;<br/><a NAME="E3:63_2_1"/>
<font color="Maroon" title="c3">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:63_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:()"><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="63_2_2"><span class="kw">suppose </span></a><a NAME="E1:63_2_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:63_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>;<br/><a NAME="E4:63_2_2"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E5:63_2_2"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_3"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:63_2_3"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<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="E4:63_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_3"><span class="lab"><font color="Green" title="E42">A5</font></span></a></span>;<br/><a NAME="E6:63_2_3"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E7:63_2_3"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_4"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:63_2_4"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<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="E4:63_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_4"><span class="lab"><font color="Green" title="E42">A6</font></span></a></span>;<br/><a NAME="E6:63_2_4"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E7:63_2_4"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_5"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:63_2_5"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<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="E4:63_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_5"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>;<br/><a NAME="E6:63_2_5"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E7:63_2_5"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_6"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:63_2_6"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<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="E4:63_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D3" title="TOPREALB:def.3">TOPREALB:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_6"><span class="lab"><font color="Green" title="E42">A8</font></span></a></span>;<br/><a NAME="E6:63_2_6"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E7:63_2_6"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_7"><span class="kw">suppose </span></a><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#T8" title="TOPALG_2:th.8">TOPALG_2:8</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><a NAME="E3:63_2_7"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E4:63_2_7"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_8"><span class="kw">suppose </span></a><a NAME="E1:63_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#NR3" title="XXREAL_0:NR.3">&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#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></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="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#T10" title="TOPALG_2:th.10">TOPALG_2:10</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/><a NAME="E3:63_2_8"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E4:63_2_8"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_9"><span class="kw">suppose </span></a><a NAME="E1:63_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#NR3" title="XXREAL_0:NR.3">&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#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> )
 ; <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="c1">X</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#T11" title="TOPALG_2:th.11">TOPALG_2:11</a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/><a NAME="E3:63_2_9"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E4:63_2_9"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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="63_2_10"><span class="kw">suppose </span></a><a NAME="E1:63_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#NR3" title="XXREAL_0:NR.3">&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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_2.html#T9" title="TOPALG_2:th.9">TOPALG_2:9</a>, <a class="ref" href="xxreal_1.html#T33" title="XXREAL_1:th.33">XXREAL_1:33</a></span>;<br/><a NAME="E3:63_2_10"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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/><span class="kw">hence </span><a NAME="E4:63_2_10"/>
<a href="topalg_2.html#K2" title="TOPALG_2:func.2">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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><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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