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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">A</font> =  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">B</font> =  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Maroon" title="c2">y</font> as   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T54" target="_self" title="TOPGEN_5:th.54">Th58</a>, <a class="ref" href="topgen_5.html#T56" target="_self" title="TOPGEN_5:th.56">Th60</a></span>;<br/>
<a NAME="E2:63"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> <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="topgen_3.html#D2" title="TOPGEN_3:def.2">TOPGEN_3:def 2</a></span>;<br/>
<span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 

<br/>.= 
<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="c1">x</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#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T382" title="XXREAL_1:th.382">XXREAL_1:382</a></span>
<br/>.= 
<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="c1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a>, <a class="ref" href="xxreal_1.html#T294" title="XXREAL_1:th.294">XXREAL_1:294</a></span>
<br/>.= 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T224" title="XXREAL_1:th.224">XXREAL_1:224</a>, <a class="ref" href="xxreal_1.html#T294" title="XXREAL_1:th.294">XXREAL_1:294</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:63"/>
<span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> is   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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