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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">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>  holds  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Olive" title="b1">C</font></span><br/><span class="kw">let </span><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#V2" title="COMPTS_1:attr.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C</font></span><br/>



<a NAME="E1:16"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></span>;<br/>
<a NAME="E2:16"/>
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-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#K6" title="PSCOMP_1:func.6">W-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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T33" title="SPRECT_1:th.33">SPRECT_1:33</a></span>;<br/>
<a NAME="E3:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-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#K6" title="PSCOMP_1:func.6">W-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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E4:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-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#K6" title="PSCOMP_1:func.6">W-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#K15" title="NEWTON:func.15">|^</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#K6" title="PSCOMP_1:func.6">W-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#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/>
<a NAME="E5:16"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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#K6" title="PSCOMP_1:func.6">W-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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-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#K6" title="PSCOMP_1:func.6">W-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#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1a.html#T83" title="JORDAN1A:th.83">JORDAN1A:83</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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