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<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span> 
 ;<br/>

<b>hence </b><a NAME="E2:29"/>
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_5.html#T2" target="_self">Th2</a></i>;<br/>

<a NAME="E3:29"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:29"/>
<a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E5:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T5">RFUNCT_2:5</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D26">PSCOMP_1:def 26</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/>
<a NAME="E8:29"/><b>then </b>
 <a href="pscomp_1.html#K11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D21">PSCOMP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E9:29"/>
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="pscomp_1.html#D32">PSCOMP_1:def 32</a></i>;<br/>

<a NAME="E10:29"/>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T5">RFUNCT_2:5</a></i>;<br/>
<a NAME="E12:29"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D26">PSCOMP_1:def 26</a></i>;<br/>
<a NAME="E13:29"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/>
<a NAME="E14:29"/><b>then </b>
 <a href="pscomp_1.html#K10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pscomp_1.html#D20">PSCOMP_1:def 20</a></i>;<br/>
<b>hence </b><a NAME="E15:29"/>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="pscomp_1.html#D33">PSCOMP_1:def 33</a></i>;<br/>

<a NAME="E16:29"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:29"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
 <a href="pscomp_1.html#K11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pscomp_1.html#D21">PSCOMP_1:def 21</a></i>;<br/>
<b>hence </b><a NAME="E18:29"/>
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="pscomp_1.html#D31">PSCOMP_1:def 31</a></i>;<br/>


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