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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,1)</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,1)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Olive" title="b1">C</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">a</font> =  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">s</font> =  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">w</font> =  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">e</font> =  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1);<br/>
<span class="kw">assume </span><a NAME="E1:51"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span><br/>

<a NAME="E2:51"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="jordan1a.html#E42"><span class="lab"><font color="Green" title="E24">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" onmouseover="rs('jordan8/D1')" onmouseout="rh()">JORDAN8:def 1</a></span>
<br/>.= 
<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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1a.html#K1" title="JORDAN1A:func.1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,1)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" onmouseover="rs('euclid/T52')" onmouseout="rh()">EUCLID:52</a></span>
<br/>.= 
<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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="jordan1a.html#E44"><span class="lab"><font color="Green" title="E26">Lm6</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2

;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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