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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="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><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">p</font> being   <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>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <font color="Olive" title="b2">x</font> &amp; 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<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="c2">n</font>)</span>)</span> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<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="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">p</font> being   <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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <font color="Olive" title="b1">x</font> &amp; 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <font color="Maroon" title="c4">x</font> &amp; 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f</font> =  <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:96"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E2:96"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E3:96"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 <span class="kw">and </span><br/><a NAME="E4:96"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:96"/><span class="lab"><font color="Green" title="E65">A5</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> </span><br/>

<span class="kw">assume </span><a NAME="E6:96"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
 not  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:96"/><span class="lab"><font color="Green" title="E67">A7</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A8</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E9:96"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:96"/><span class="kw">then </span>
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:96"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A9</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:96"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E13:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A10</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:96"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="sppol_2.html#T17" title="SPPOL_2:th.17">SPPOL_2:17</a></span>;<br/>
<a NAME="E15:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A11</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:96"/><span class="lab"><font color="Green" title="E72">A12</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>) <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan9.html#D1" title="JORDAN9:def.1">JORDAN9:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A13</font></span>: <font color="Maroon" title="c4">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="ref" href="sppol_1.html#T19" title="SPPOL_1:th.19">SPPOL_1:19</a>, <a class="ref" href="sppol_1.html#T40" title="SPPOL_1:th.40">SPPOL_1:40</a></span>
;<br/>
<span class="lab"><font color="Green" title="E74">A14</font></span>: <font color="Maroon" title="c4">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:96"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="txt" href="#E6:96"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="ref" href="sppol_1.html#T19" title="SPPOL_1:th.19">SPPOL_1:19</a>, <a class="ref" href="sppol_1.html#T40" title="SPPOL_1:th.40">SPPOL_1:40</a></span>
;<br/>
<a NAME="E19:96"/><span class="lab"><font color="Green" title="E75">A15</font></span>: 
 <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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E20:96"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A16</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/>
<a NAME="E21:96"/><span class="lab"><font color="Green" title="E77">A17</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:96_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_3_1"><span class="kw">suppose </span></a><a NAME="E1:96_3_1"/><span class="lab"><font color="Green" title="E78">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:96_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="topreal1.html#T3" title="TOPREAL1:th.3">TOPREAL1:3</a></span>;<br/><a NAME="E3:96_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E77">A17</font></span></a>, <a class="txt" href="#E15:96"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="jordan1a.html#T75" target="_self" title="JORDAN1A:th.75">Th96</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:96_3_1"/><span class="lab"><font color="Green" title="E80">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</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_1.html#K2" title="MATRIX_1: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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:96_3_1"/><span class="lab"><font color="Green" title="E81">A21</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E7:96_3_1"/><span class="lab"><font color="Green" title="E82">A22</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:96_3_1"/><span class="lab"><font color="Green" title="E83">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j1</font>,<font color="Maroon" title="c11">j2</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_1.html#K2" title="MATRIX_1: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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:96_3_1"/><span class="lab"><font color="Green" title="E84">A24</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c10">j1</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E11:96_3_1"/><span class="lab"><font color="Green" title="E85">A25</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E12:96_3_1"/><span class="lab"><font color="Green" title="E86">A26</font></span>: 
<font color="Maroon" title="c8">i1</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:96_3_1"/><span class="lab"><font color="Green" title="E87">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:96_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A28</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:96"><span class="lab"><font color="Green" title="E74">A14</font></span></a>, <a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/><a NAME="E3:96_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A29</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="txt" href="#E6:96_3_1"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="txt" href="#E9:96_3_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="txt" href="#E10:96_3_1"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:96_3_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="txt" href="#E5:96_3_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="txt" href="#E6:96_3_1"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="txt" href="#E9:96_3_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="txt" href="#E10:96_3_1"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E2:96_3_1_1"><span class="lab"><font color="Green" title="E88">A28</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><span class="kw">then </span>1 = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">j2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T2" title="GOBOARD7:th.2">GOBOARD7:2</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_3_1_1"><span class="lab"><font color="Green" title="E89">A29</font></span></a>, <a class="ref" href="goboard7.html#T2" title="GOBOARD7:th.2">GOBOARD7:2</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:96_3_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:96_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_3_1"><span class="lab"><font color="Green" title="E78">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:96_3_1"/><span class="lab"><font color="Green" title="E89">A31</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:96_3_1"/>
<font color="Maroon" title="c10">j1</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:96_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_3_1"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="txt" href="#E7:96_3_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="txt" href="#E13:96_3_1"><span class="lab"><font color="Green" title="E87">A27</font></span></a>, <a class="txt" href="#E10:96_3_1"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E11:96_3_1"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E15:96_3_1"><span class="lab"><font color="Green" title="E88">A30</font></span></a>, <a class="ref" href="jordan1a.html#T18" target="_self" title="JORDAN1A:th.18">Th39</a></span>;<br/><a NAME="E19:96_3_1"/><span class="kw">then </span>
 <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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_3_1"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:96_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A32</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pscomp_1.html#T47" title="PSCOMP_1:th.47">PSCOMP_1:47</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c3">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<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="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E75">A15</font></span></a>, <a class="txt" href="#E7:96_3_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>
;<br/><a NAME="E22:96_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E75">A15</font></span></a>, <a class="txt" href="#E20:96"><span class="lab"><font color="Green" title="E76">A16</font></span></a>, <a class="txt" href="#E7:96_3_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="txt" href="#E10:96_3_1"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="txt" href="#E11:96_3_1"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E16:96_3_1"><span class="lab"><font color="Green" title="E89">A31</font></span></a>, <a class="txt" href="#E20:96_3_1"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="ref" href="jordan1a.html#T18" target="_self" title="JORDAN1A:th.18">Th39</a></span>;<br/><a NAME="E23:96_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="txt" href="#E18:96"><span class="lab"><font color="Green" title="E74">A14</font></span></a>, <a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="txt" href="#E3:96_3_1"><span class="lab"><font color="Green" title="E79">A19</font></span></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a>, <a class="ref" href="sppol_2.html#T17" title="SPPOL_2:th.17">SPPOL_2:17</a></span>;<br/><a NAME="E24:96_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E77">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E25:96_3_1"/><span class="kw">then </span>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E26:96_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan10.html#T5" title="JORDAN10:th.5">JORDAN10:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_3_2"><span class="kw">suppose </span></a><a NAME="E1:96_3_2"/><span class="lab"><font color="Green" title="E78">A33</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:96_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="topreal1.html#T3" title="TOPREAL1:th.3">TOPREAL1:3</a></span>;<br/><a NAME="E3:96_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A34</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E77">A17</font></span></a>, <a class="txt" href="#E15:96"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="jordan1a.html#T75" target="_self" title="JORDAN1A:th.75">Th96</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:96_3_2"/><span class="lab"><font color="Green" title="E80">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</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_1.html#K2" title="MATRIX_1: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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:96_3_2"/><span class="lab"><font color="Green" title="E81">A36</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E7:96_3_2"/><span class="lab"><font color="Green" title="E82">A37</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:96_3_2"/><span class="lab"><font color="Green" title="E83">A38</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">j1</font>,<font color="Maroon" title="c11">j2</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_1.html#K2" title="MATRIX_1: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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:96_3_2"/><span class="lab"><font color="Green" title="E84">A39</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c10">j1</font>,<font color="Maroon" title="c11">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><a NAME="E11:96_3_2"/><span class="lab"><font color="Green" title="E85">A40</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E12:96_3_2"/><span class="lab"><font color="Green" title="E86">A41</font></span>: 
<font color="Maroon" title="c8">i1</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:96_3_2"/><span class="lab"><font color="Green" title="E87">A42</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:96_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:96_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A43</font></span>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:96"><span class="lab"><font color="Green" title="E74">A14</font></span></a>, <a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/><a NAME="E3:96_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A44</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_3_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E6:96_3_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E9:96_3_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E10:96_3_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:96_3_2_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">i1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:96"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E13:96"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="txt" href="#E11:96"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="txt" href="#E5:96_3_2"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E6:96_3_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E9:96_3_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E10:96_3_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="#E2:96_3_2_1"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>;<br/><span class="kw">then </span>1 = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">j2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_3_2_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="ref" href="goboard7.html#T2" title="GOBOARD7:th.2">GOBOARD7:2</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96_3_2_1"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="ref" href="goboard7.html#T2" title="GOBOARD7:th.2">GOBOARD7:2</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:96_3_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:96_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A45</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_3_2"><span class="lab"><font color="Green" title="E78">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:96_3_2"/><span class="lab"><font color="Green" title="E89">A46</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:96_3_2"/>
<font color="Maroon" title="c10">j1</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:96_3_2"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:96_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:96_3_2"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E7:96_3_2"><span class="lab"><font color="Green" title="E82">A37</font></span></a>, <a class="txt" href="#E13:96_3_2"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E10:96_3_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="#E11:96_3_2"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E15:96_3_2"><span class="lab"><font color="Green" title="E88">A45</font></span></a>, <a class="ref" href="jordan1a.html#T18" target="_self" title="JORDAN1A:th.18">Th39</a></span>;<br/><a NAME="E19:96_3_2"/><span class="kw">then </span>
 <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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_3_2"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:96_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A47</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pscomp_1.html#T47" title="PSCOMP_1:th.47">PSCOMP_1:47</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c3">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<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="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E75">A15</font></span></a>, <a class="txt" href="#E7:96_3_2"><span class="lab"><font color="Green" title="E82">A37</font></span></a>, <a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>
;<br/><a NAME="E22:96_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:96"><span class="lab"><font color="Green" title="E75">A15</font></span></a>, <a class="txt" href="#E20:96"><span class="lab"><font color="Green" title="E76">A16</font></span></a>, <a class="txt" href="#E7:96_3_2"><span class="lab"><font color="Green" title="E82">A37</font></span></a>, <a class="txt" href="#E10:96_3_2"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="#E11:96_3_2"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E16:96_3_2"><span class="lab"><font color="Green" title="E89">A46</font></span></a>, <a class="txt" href="#E20:96_3_2"><span class="lab"><font color="Green" title="E90">A47</font></span></a>, <a class="ref" href="jordan1a.html#T18" target="_self" title="JORDAN1A:th.18">Th39</a></span>;<br/><a NAME="E23:96_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="txt" href="#E18:96"><span class="lab"><font color="Green" title="E74">A14</font></span></a>, <a class="txt" href="#E17:96"><span class="lab"><font color="Green" title="E73">A13</font></span></a>, <a class="txt" href="#E3:96_3_2"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a>, <a class="ref" href="sppol_2.html#T17" title="SPPOL_2:th.17">SPPOL_2:17</a></span>;<br/><a NAME="E24:96_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:96"><span class="lab"><font color="Green" title="E77">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E25:96_3_2"/><span class="kw">then </span>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E26:96_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan10.html#T5" title="JORDAN10:th.5">JORDAN10:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
