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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E74">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">p'</font> = <font color="Maroon" title="c1">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E3:115"/><i><font color="Green" title="E75">A2</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:115"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T69" target="_self" title="PSCOMP_1:th.69">Th69</a></i>;<br/>
<b>hence </b><a NAME="E4:115"/>
 <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T47" target="_self" title="PSCOMP_1:th.47">Th47</a></i>;<br/>

<b>thus </b><a NAME="E5:115"/>
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:115"><i><font color="Green" title="E75">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T50" target="_self" title="PSCOMP_1:th.50">Th50</a></i>;<br/>

<a NAME="E6:115"/><i><font color="Green" title="E76">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:115"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T70" target="_self" title="PSCOMP_1:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E7:115"/>
 <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T47" target="_self" title="PSCOMP_1:th.47">Th47</a></i>;<br/>

<b>thus </b><a NAME="E8:115"/>
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:115"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="pscomp_1.html#T50" target="_self" title="PSCOMP_1:th.50">Th50</a></i>;<br/>


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