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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> non <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> non <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</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/>



<a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E2:12"/>
 <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="sprect_1.html#T34" title="SPRECT_1:th.34">SPRECT_1:34</a></i>;<br/>
<a NAME="E3:12"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan19.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E5:12"/>
 <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan10.html#T6" title="JORDAN10:th.6">JORDAN10:6</a></i>;<br/>
<b>hence </b><a NAME="E6:12"/>
 <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c2">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E4:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>


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