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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

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



<b>assume </b><a NAME="E2:11"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>assume </b><a NAME="E6:11"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:11"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E9:11"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>assume </b><a NAME="E10:11"/>
not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:11"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E13:11"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font>, <font color="Maroon">s</font> = <font color="Maroon">s</font>, <font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">P1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Q1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="seq_4.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<font color="Maroon">Y</font> <a href="subset_1.html#K3">`</a>  = 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">t</font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="limfunc1.html#T25">LIMFUNC1:25</a></i>
;<br/>
<b>then </b><i><font color="Green">E85</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T175">RELAT_1:175</a></i>
;<br/>
<a NAME="E18:11"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T50">FUNCT_2:50</a></i>;<br/>
<a NAME="E19:11"/>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E20:11"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T141">FUNCT_1:141</a></i>;<br/>
<a NAME="E21:11"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E22:11"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<b>then </b><i><font color="Green">E87</font></i>: <font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E24:11"/><i><font color="Green">E88</font></i>: 
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/>
<a NAME="E25:11"/><i><font color="Green">E90</font></i>: 
 <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E26:11"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E27:11"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E28:11"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E29:11"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E30:11"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E31:11"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E32:11"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E33:11"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E34:11"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E35:11"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E36:11"/>
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="fcont_3.html#T8">FCONT_3:8</a></i>;<br/>
<a NAME="E37:11"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E38:11"/>
 <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="fcont_3.html#T7">FCONT_3:7</a></i>;<br/>
<a NAME="E39:11"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T54">PSCOMP_1:54</a></i>;<br/>
<a NAME="E40:11"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E41:11"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">t</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T18">LIMFUNC1:18</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>
;<br/>
<a NAME="E43:11"/><b>then </b>
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font> <a href="xboole_0.html#R1">misses</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E44:11"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T141">FUNCT_1:141</a></i>;<br/>
<a NAME="E45:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E46:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/>
<a NAME="E47:11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i>, , , , <a class="ref" href="tsep_1.html#T49">TSEP_1:49</a></i>;<br/>
<b>hence </b><a NAME="E48:11"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/>


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