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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">r</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" target="_self" title="JCT_MISC:def.1">JCT_MISC:def 1</a><br/><b>let </b><font color="Maroon" title="c5">s</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> ;<br/>



<b>assume </b><a NAME="E2:11"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>assume </b><a NAME="E6:11"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:11"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>and </b><br/><a NAME="E9:11"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>assume </b><a NAME="E10:11"/>
not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</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">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E12:11"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E13:11"/><i><font color="Green" title="E16">A9</font></i>: 
not <font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c8">t</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">P1</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">Q1</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">P</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>;<br/>
<b>set </b><font color="Maroon" title="c15">Q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>;<br/>
<b>set </b><font color="Maroon" title="c16">X</font> = <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Y</font> = <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">t</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> as   <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>  ;<br/>
<font color="Maroon" title="c17">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c11">t</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
<b>by </b><i><a class="ref" href="limfunc1.html#T25" title="LIMFUNC1:th.25">LIMFUNC1:25</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E17">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c17">Y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jct_misc.html#T5" target="_self" title="JCT_MISC:th.5">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T175" title="RELAT_1:th.175">RELAT_1:175</a></i>
;<br/>
<a NAME="E18:11"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T50" title="FUNCT_2:th.50">FUNCT_2:50</a></i>;<br/>
<a NAME="E19:11"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">t</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E13:11"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="funct_1.html#T141" title="FUNCT_1:th.141">FUNCT_1:141</a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E20:11"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c11">t</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E18:11"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E21:11"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E17:11"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<b>then </b><i><font color="Green" title="E19">A12</font></i>: <font color="Maroon" title="c3">A</font> = 
<font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E23:11"/><i><font color="Green" title="E20">A13</font></i>: 
 <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">t</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T151" title="LIMFUNC1:th.151">LIMFUNC1:151</a></i>;<br/>
<a NAME="E24:11"/><i><font color="Green" title="E21">A14</font></i>: 
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>
<a NAME="E25:11"/>
( <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">t</font> &amp; <font color="Maroon" title="c9">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">t</font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="jct_misc.html#E12:11"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="jct_misc.html#E13:11"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:11"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E27:11"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E23:11"><i><font color="Green" title="E20">A13</font></i></a></i>;<br/>
<a NAME="E28:11"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E5:11"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E29:11"/><b>then </b><i><font color="Green" title="E22">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E4:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:11"/>
( <font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">t</font> )
 
<b>by </b><i><a class="txt" href="jct_misc.html#E6:11"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="jct_misc.html#E12:11"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="jct_misc.html#E13:11"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E31:11"/><b>then </b>
<font color="Maroon" title="c11">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">s</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E32:11"/><b>then </b>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E24:11"><i><font color="Green" title="E21">A14</font></i></a></i>;<br/>
<a NAME="E33:11"/><b>then </b>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E9:11"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E34:11"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="jct_misc.html#E8:11"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:11"/>
 <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="fcont_3.html#T8" title="FCONT_3:th.8">FCONT_3:8</a></i>;<br/>
<a NAME="E36:11"/><b>then </b><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T54" title="PSCOMP_1:th.54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E37:11"/>
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="fcont_3.html#T7" title="FCONT_3:th.7">FCONT_3:7</a></i>;<br/>
<a NAME="E38:11"/><b>then </b><i><font color="Green" title="E25">A18</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T54" title="PSCOMP_1:th.54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E39:11"/><i><font color="Green" title="E26">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E40:11"/><i><font color="Green" title="E27">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c11">t</font>,<font color="Maroon" title="c11">t</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T18" title="LIMFUNC1:th.18">LIMFUNC1:18</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="rcomp_1.html#T12" title="RCOMP_1:th.12">RCOMP_1:12</a></i>
;<br/>
<a NAME="E42:11"/><b>then </b>
 <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E43:11"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T141" title="FUNCT_1:th.141">FUNCT_1:141</a></i>;<br/>
<a NAME="E44:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E45:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T65" title="XBOOLE_1:th.65">XBOOLE_1:65</a></i>;<br/>
<a NAME="E46:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="jct_misc.html#E36:11"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="jct_misc.html#E38:11"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="jct_misc.html#E39:11"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="jct_misc.html#E40:11"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="tsep_1.html#T49" title="TSEP_1:th.49">TSEP_1:49</a></i>;<br/>
<b>hence </b><a NAME="E47:11"/>
contradiction
 <b>by </b><i><a class="txt" href="jct_misc.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="jct_misc.html#E22:11"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="jct_misc.html#E29:11"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="jct_misc.html#E34:11"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="connsp_1.html#T16" title="CONNSP_1:th.16">CONNSP_1:16</a></i>;<br/>


</div>
