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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font>, <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</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="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span>)</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span>)</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">k</font>)</span>)</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="Olive" title="b1">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span>) <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Olive" title="b1">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span>)</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span>)</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span>) <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font>, <font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>) <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c6">Ga</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">US</font> =  <a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">LS</font> =  <a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">UA</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">Wmin</font> =  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Emax</font> =  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Wbo</font> =  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">Ebo</font> =  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Gik</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Gij</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c3">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">and </span><br/><a NAME="E5:16"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c4">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E7:16"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>) <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font>
 ; <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="E9:16"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E19">A7</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="E11:16"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E12:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</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="E13:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan1j.html#T57" title="JORDAN1J:th.57">JORDAN1J:57</a></span>;<br/>
<a NAME="E14:16"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:16"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c5">k</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:16"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<a NAME="E17:16"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">do</font> =  <a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c17">go</font> =  <a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>);<br/>
<a NAME="E18:16"/><span class="lab"><font color="Green" title="E28">A16</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="c2">C</font>,<font color="Maroon" title="c1">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="c2">C</font>,<font color="Maroon" title="c1">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="E19:16"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#T15" title="JORDAN1E:th.15">JORDAN1E:15</a></span>;<br/>
<a NAME="E20:16"/><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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:16"/><span class="kw">then </span>
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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A18</font></span>: <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T5" title="JORDAN1F:th.5">JORDAN1F:5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E31">A19</font></span>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">k</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="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T73" title="JORDAN1A:th.73">JORDAN1A:73</a></span>
;<br/>
<a NAME="E24:16"/>
 <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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T10" title="JORDAN8:th.10">JORDAN8:10</a></span>;<br/>
<a NAME="E25:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A20</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E26:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c5">k</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<a NAME="E27:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">go</font> =  <a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>) as   <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="jordan3.html#T35" title="JORDAN3:th.35">JORDAN3:35</a></span>;<br/>
<a NAME="E29:16"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#T15" title="JORDAN1E:th.15">JORDAN1E:15</a></span>;<br/>
<a NAME="E30:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E31:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A25</font></span>: 
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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E32:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T8" title="JORDAN1F:th.8">JORDAN1F:8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E39">A27</font></span>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">k</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="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T73" title="JORDAN1A:th.73">JORDAN1A:73</a></span>
;<br/>
<a NAME="E35:16"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<a NAME="E36:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A29</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E33:16"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">do</font> =  <a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>) as   <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan3.html#T34" title="JORDAN3:th.34">JORDAN3:34</a></span>;<br/>
<a NAME="E38:16"/><span class="lab"><font color="Green" title="E42">A30</font></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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c5">k</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<span class="lab"><font color="Green" title="E43">A31</font></span>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T6" title="JORDAN1F:th.6">JORDAN1F:6</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c5">k</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="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>
;<br/>
<a NAME="E41:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A32</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E35:16"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E39:16"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></span>;<br/>
<a NAME="E42:16"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">go</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E43:16"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="jordan1g.html#T4" title="JORDAN1G:th.4">JORDAN1G:4</a>, <a class="ref" href="jordan1j.html#T40" title="JORDAN1J:th.40">JORDAN1J:40</a></span>;<br/>
<a NAME="E44:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c18">go</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1g.html#T4" title="JORDAN1G:th.4">JORDAN1G:4</a>, <a class="ref" href="jordan1j.html#T38" title="JORDAN1J:th.38">JORDAN1J:38</a></span>;<br/>
<a NAME="E45:16"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E46:16"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="jordan1g.html#T5" title="JORDAN1G:th.5">JORDAN1G:5</a>, <a class="ref" href="jordan1j.html#T40" title="JORDAN1J:th.40">JORDAN1J:40</a></span>;<br/>
<a NAME="E47:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A38</font></span>: 
<font color="Maroon" title="c19">do</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1g.html#T5" title="JORDAN1G:th.5">JORDAN1G:5</a>, <a class="ref" href="jordan1j.html#T39" title="JORDAN1J:th.39">JORDAN1J:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">go</font> = <font color="Maroon" title="c18">go</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>   <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="jgraph_1.html#T12" title="JGRAPH_1:th.12">JGRAPH_1:12</a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">do</font> = <font color="Maroon" title="c19">do</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>   <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="jgraph_1.html#T12" title="JGRAPH_1:th.12">JGRAPH_1:12</a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></span>;<br/>
<a NAME="E50:16"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E51:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A41</font></span>: <font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> = 
<font color="Maroon" title="c20">go</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="jordan3.html#T24" title="JORDAN3:th.24">JORDAN3:24</a></span>
;<br/>
<a NAME="E53:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E54:16"/><span class="kw">then </span>
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> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E54">A42</font></span>: <font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c21">do</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan3.html#T23" title="JORDAN3:th.23">JORDAN3:23</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:16"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="ref" href="finseq_3.html#T26" title="FINSEQ_3:th.26">FINSEQ_3:26</a></span>;<br/>
<a NAME="E57:16"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<font color="Maroon" title="c22">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font>
 
;<br/>
<a NAME="E58:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A44</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E59:16"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T19" title="TOPREAL3:th.19">TOPREAL3:19</a></span>;<br/>
<a NAME="E60:16"/><span class="lab"><font color="Green" title="E58">A46</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></span>;<br/>
<a NAME="E61:16"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:16"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E62:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E63:16"/>
<font color="Maroon" title="c22">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E64:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A48</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</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"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">m</font></span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E57:16"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E65:16"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_9"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span></span><br/>

<a NAME="E2:16_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A49</font></span>: 
<font color="Maroon" title="c23">x</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_9"/><span class="lab"><font color="Green" title="E62">A50</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:16"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E4:16_9"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_9"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_9"><span class="lab"><font color="Green" title="E61">A49</font></span></a>, <a class="txt" href="#E3:16_9"><span class="lab"><font color="Green" title="E62">A50</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E66:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A51</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:16"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E67:16"/><span class="lab"><font color="Green" title="E62">A52</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T19" title="TOPREAL3:th.19">TOPREAL3:19</a></span>;<br/>
<a NAME="E68:16"/><span class="lab"><font color="Green" title="E63">A53</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan3.html#T42" title="JORDAN3:th.42">JORDAN3:42</a></span>;<br/>
<a NAME="E69:16"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:16"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E70:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A54</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E71:16"/><span class="lab"><font color="Green" title="E65">A55</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1) <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <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="#E45:16"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E72:16"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span></span><br/>

<a NAME="E2:16_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c23">x</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_10"/><span class="lab"><font color="Green" title="E67">A57</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:16"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E4:16_10"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_10"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_10"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="txt" href="#E3:16_10"><span class="lab"><font color="Green" title="E67">A57</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E73:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A58</font></span>: 
<span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:16"><span class="lab"><font color="Green" title="E64">A54</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="lab"><font color="Green" title="E67">A59</font></span>: <font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="sprect_3.html#T22" title="SPRECT_3:th.22">SPRECT_3:22</a></span>
<br/>.= 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T5" title="JORDAN1F:th.5">JORDAN1F:5</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E68">A60</font></span>: <font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T8" title="JORDAN1F:th.8">JORDAN1F:8</a></span>
<br/>.= 
<font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">do</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan1j.html#T35" title="JORDAN1J:th.35">JORDAN1J:35</a></span>
;<br/>
<a NAME="E76:16"/><span class="lab"><font color="Green" title="E69">A61</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">go</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a></span>;<br/>
<a NAME="E77:16"/><span class="lab"><font color="Green" title="E70">A62</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c21">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:16"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a></span>;<br/>
<a NAME="E78:16"/><span class="lab"><font color="Green" title="E71">A63</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c23">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_13"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c23">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span></span><br/>

<a NAME="E2:16_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A64</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T42" title="FINSEQ_6:th.42">FINSEQ_6:42</a></span>;<br/>
<a NAME="E4:16_13"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:16"><span class="lab"><font color="Green" title="E68">A60</font></span></a>, <a class="txt" href="#E2:16_13"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="ref" href="revrot_1.html#T3" title="REVROT_1:th.3">REVROT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_13"/>
<font color="Maroon" title="c23">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:16"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E77:16"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="txt" href="#E3:16_13"><span class="lab"><font color="Green" title="E73">A65</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<span class="lab"><font color="Green" title="E72">A66</font></span>: <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T6" title="JORDAN1F:th.6">JORDAN1F:6</a></span>
;<br/>
<a NAME="E80:16"/><span class="lab"><font color="Green" title="E73">A67</font></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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/>
<a NAME="E81:16"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_15"/><span class="lab"><font color="Green" title="E74">A68</font></span>: 
<font color="Maroon" title="c23">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:16_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A69</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:16_15"/><span class="lab"><font color="Green" title="E76">A70</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_15"><span class="lab"><font color="Green" title="E74">A68</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="E4:16_15"/><span class="kw">then </span>
<font color="Maroon" title="c23">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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E68:16"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E2:16_15"><span class="lab"><font color="Green" title="E75">A69</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="E5:16_15"/><span class="kw">then </span>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#T16" title="JORDAN1E:th.16">JORDAN1E:16</a></span>;<br/>
<a NAME="E6:16_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A71</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_15_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_15_1"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 ; <a 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="E2:16_15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A72</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E79:16"><span class="lab"><font color="Green" title="E72">A66</font></span></a>, <a class="txt" href="#E3:16_15"><span class="lab"><font color="Green" title="E76">A70</font></span></a>, <a class="ref" href="jordan1e.html#T7" title="JORDAN1E:th.7">JORDAN1E:7</a></span>;<br/><a NAME="E3:16_15_1"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/><a NAME="E4:16_15_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E80:16"><span class="lab"><font color="Green" title="E73">A67</font></span></a>, <a class="txt" href="#E2:16_15_1"><span class="lab"><font color="Green" title="E78">A72</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:16_15_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</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>
<span class="kw">hence </span><a NAME="E8:16_15"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:16"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="txt" href="#E6:16_15"><span class="lab"><font color="Green" title="E77">A71</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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>
<a NAME="E82:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A73</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c21">do</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:16"><span class="lab"><font color="Green" title="E71">A63</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">W2</font> = <font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2;<br/>
<a NAME="E83:16"/><span class="lab"><font color="Green" title="E75">A74</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E76">A75</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_16"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_16"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 ; <a 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="E2:16_16"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c5">k</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T73" title="JORDAN1A:th.73">JORDAN1A:73</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</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>
<font color="Maroon" title="c20">go</font> = 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="jordan1g.html#T49" title="JORDAN1G:th.49">JORDAN1G:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a>, <a class="ref" href="finseq_6.html#T116" title="FINSEQ_6:th.116">FINSEQ_6:116</a></span>
;<br/>
<a NAME="E86:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A76</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:16"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/>
<a NAME="E87:16"/><span class="lab"><font color="Green" title="E78">A77</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:16"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="ref" href="finseq_6.html#T42" title="FINSEQ_6:th.42">FINSEQ_6:42</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">pion</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E79">A78</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_18"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">n</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:16_18"/>
<font color="Maroon" title="c25">n</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:16_18"/><span class="kw">then </span>
<font color="Maroon" title="c25">n</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> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/><a NAME="E3:16_18"/><span class="kw">then </span>
( <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_18"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</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>
<a NAME="E89:16"/><span class="lab"><font color="Green" title="E80">A79</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/>
<span class="lab"><font color="Green" title="E81">A80</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,1)</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="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/>
<a NAME="E91:16"/><span class="kw">then </span>
 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>) is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_1.html#T16" title="SPPOL_1:th.16">SPPOL_1:16</a></span>;<br/>
<a NAME="E92:16"/><span class="kw">then </span>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:16"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="ref" href="jordan1b.html#T7" title="JORDAN1B:th.7">JORDAN1B:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">pion1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E94:16"/><span class="lab"><font color="Green" title="E82">A81</font></span>: 
<font color="Maroon" title="c25">pion1</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E95:16"/><span class="lab"><font color="Green" title="E83">A82</font></span>: 
<font color="Maroon" title="c25">pion1</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E96:16"/><span class="lab"><font color="Green" title="E84">A83</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c25">pion1</font>
 <span class="kw">and </span><br/><a NAME="E97:16"/><span class="lab"><font color="Green" title="E85">A84</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">and </span><br/><a NAME="E98:16"/><span class="lab"><font color="Green" title="E86">A85</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">pion1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E99:16"/><span class="lab"><font color="Green" title="E87">A86</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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> <font color="Maroon" title="c25">pion1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:16"><span class="lab"><font color="Green" title="E79">A78</font></span></a>, <a class="ref" href="goboard3.html#T2" title="GOBOARD3:th.2">GOBOARD3:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">pion1</font> = <font color="Maroon" title="c25">pion1</font> as   <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E95:16"><span class="lab"><font color="Green" title="E83">A82</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">godo</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c21">do</font>;<br/>
<a NAME="E101:16"/><span class="lab"><font color="Green" title="E88">A87</font></span>: 
1 <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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T34" title="GOBOARD7:th.34">GOBOARD7:34</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E102:16"/><span class="lab"><font color="Green" title="E89">A88</font></span>: 
1 <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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T34" title="GOBOARD7:th.34">GOBOARD7:34</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E103:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T7" title="TOPREAL8:th.7">TOPREAL8:7</a></span>;<br/>
<a NAME="E104:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A89</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E105:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A90</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E106:16"/><span class="lab"><font color="Green" title="E92">A91</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c21">do</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T7" title="TOPREAL8:th.7">TOPREAL8:7</a></span>;<br/>
<a NAME="E107:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A92</font></span>: 
1 <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"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c21">do</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:16"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E108:16"/><span class="lab"><font color="Green" title="E94">A93</font></span>: 
 <a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1g.html#T4" title="JORDAN1G:th.4">JORDAN1G:4</a></span>;<br/>
<a NAME="E109:16"/><span class="lab"><font color="Green" title="E95">A94</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E97:16"><span class="lab"><font color="Green" title="E85">A84</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/>
<a NAME="E110:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A95</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="txt" href="#E94:16"><span class="lab"><font color="Green" title="E82">A81</font></span></a>, <a class="ref" href="topreal8.html#T12" title="TOPREAL8:th.12">TOPREAL8:12</a></span>;<br/>
<span class="lab"><font color="Green" title="E97">A96</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="ref" href="graph_2.html#T54" title="GRAPH_2:th.54">GRAPH_2:54</a></span>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
<font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
;<br/>
<a NAME="E112:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A97</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c21">do</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:16"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="txt" href="#E110:16"><span class="lab"><font color="Green" title="E96">A95</font></span></a>, <a class="ref" href="topreal8.html#T12" title="TOPREAL8:th.12">TOPREAL8:12</a></span>;<br/>
<a NAME="E113:16"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="ref" href="topreal3.html#T19" title="TOPREAL3:th.19">TOPREAL3:19</a></span>;<br/>
<a NAME="E114:16"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</a></span>;<br/>
<a NAME="E115:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A98</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<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> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E66:16"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E116:16"/><span class="lab"><font color="Green" title="E100">A99</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="#E99:16"><span class="lab"><font color="Green" title="E87">A86</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E117:16"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_21"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span></span><br/>

<a NAME="E2:16_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A100</font></span>: 
<font color="Maroon" title="c28">x</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_21"/><span class="lab"><font color="Green" title="E102">A101</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:16"><span class="lab"><font color="Green" title="E60">A48</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E4:16_21"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="txt" href="#E116:16"><span class="lab"><font color="Green" title="E100">A99</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_21"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<font color="Maroon" title="c22">m</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#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_21"><span class="lab"><font color="Green" title="E101">A100</font></span></a>, <a class="txt" href="#E3:16_21"><span class="lab"><font color="Green" title="E102">A101</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E118:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">go</font>,<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> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E58:16"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E115:16"><span class="lab"><font color="Green" title="E99">A98</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E119:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A102</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="topreal8.html#T34" title="TOPREAL8:th.34">TOPREAL8:34</a></span>;<br/>
<a NAME="E120:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <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="#E99:16"><span class="lab"><font color="Green" title="E87">A86</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E121:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A103</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E122:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T13" title="GRAPH_2:th.13">GRAPH_2:13</a></span>;<br/>
<span class="kw">then </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"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E121:16"><span class="lab"><font color="Green" title="E102">A103</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
;<br/>
<a NAME="E124:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A104</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"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E125:16"/><span class="lab"><font color="Green" title="E104">A105</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E116:16"><span class="lab"><font color="Green" title="E100">A99</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E126:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A106</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E106">A107</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E121:16"><span class="lab"><font color="Green" title="E102">A103</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:16"><span class="lab"><font color="Green" title="E104">A105</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
;<br/>
<a NAME="E128:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <font color="Maroon" title="c26">pion1</font>
 
;<br/>
<a NAME="E129:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A108</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <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> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E126:16"><span class="lab"><font color="Green" title="E105">A106</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E130:16"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="ref" href="topreal3.html#T19" title="TOPREAL3:th.19">TOPREAL3:19</a></span>;<br/>
<a NAME="E131:16"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</a></span>;<br/>
<a NAME="E132:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A109</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:16"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E133:16"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_24"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_24"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span></span><br/>

<a NAME="E2:16_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A110</font></span>: 
<font color="Maroon" title="c28">x</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_24"/><span class="lab"><font color="Green" title="E110">A111</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:16"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="txt" href="#E126:16"><span class="lab"><font color="Green" title="E105">A106</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
;<br/>
<a NAME="E5:16_24"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:16"><span class="lab"><font color="Green" title="E104">A105</font></span></a>, <a class="txt" href="#E126:16"><span class="lab"><font color="Green" title="E105">A106</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16_24"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_24"><span class="lab"><font color="Green" title="E109">A110</font></span></a>, <a class="txt" href="#E3:16_24"><span class="lab"><font color="Green" title="E110">A111</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E134:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c26">pion1</font>,<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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E132:16"><span class="lab"><font color="Green" title="E108">A109</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E135:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A112</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span>,<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> <font color="Maroon" title="c20">go</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <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"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="txt" href="#E111:16"><span class="lab"><font color="Green" title="E97">A96</font></span></a>, <a class="txt" href="#E127:16"><span class="lab"><font color="Green" title="E106">A107</font></span></a>, <a class="txt" href="#E129:16"><span class="lab"><font color="Green" title="E107">A108</font></span></a>, <a class="ref" href="topreal8.html#T31" title="TOPREAL8:th.31">TOPREAL8:31</a></span>;<br/>
<a NAME="E136:16"/><span class="lab"><font color="Green" title="E110">A113</font></span>: 
 not <font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:16"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="ref" href="nat_d.html#T60" title="NAT_D:th.60">NAT_D:60</a></span>;<br/>
<a NAME="E137:16"/><span class="lab"><font color="Green" title="E111">A114</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">pion1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E116:16"><span class="lab"><font color="Green" title="E100">A99</font></span></a>, <a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a></span>;<br/>
<a NAME="E138:16"/><span class="lab"><font color="Green" title="E112">A115</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_25"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c28">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_25"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c28">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span></span><br/>

<a NAME="E2:16_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A116</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A117</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="revrot_1.html#T3" title="REVROT_1:th.3">REVROT_1:3</a></span>;<br/>
<a NAME="E4:16_25"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_25"><span class="lab"><font color="Green" title="E113">A116</font></span></a>, <a class="ref" href="finseq_6.html#T42" title="FINSEQ_6:th.42">FINSEQ_6:42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_25"/>
<font color="Maroon" title="c28">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:16"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E137:16"><span class="lab"><font color="Green" title="E111">A114</font></span></a>, <a class="txt" href="#E3:16_25"><span class="lab"><font color="Green" title="E114">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E139:16"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_26"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span> or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_26"/><span class="lab"><font color="Green" title="E113">A118</font></span>: 
<font color="Maroon" title="c28">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> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:16_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A119</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:16_26"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_26"><span class="lab"><font color="Green" title="E113">A118</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="E4:16_26"/><span class="kw">then </span>
<font color="Maroon" title="c28">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> <font color="Maroon" title="c26">pion1</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E2:16_26"><span class="lab"><font color="Green" title="E114">A119</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_26"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E52:16"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</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>
<a NAME="E140:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A120</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E138:16"><span class="lab"><font color="Green" title="E112">A115</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E141:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A121</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="jordan1j.html#T54" title="JORDAN1J:th.54">JORDAN1J:54</a></span>;<br/>
<a NAME="E142:16"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:16"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E137:16"><span class="lab"><font color="Green" title="E111">A114</font></span></a>, <a class="txt" href="#E140:16"><span class="lab"><font color="Green" title="E113">A120</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E143:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A122</font></span>: 
<font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1j.html#T55" title="JORDAN1J:th.55">JORDAN1J:55</a></span>;<br/>
<span class="lab"><font color="Green" title="E116">A123</font></span>: <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
<font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
;<br/>
<a NAME="E145:16"/><span class="lab"><font color="Green" title="E117">A124</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_28"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c28">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> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_28"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c28">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> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span></span><br/>

<a NAME="E2:16_28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A125</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:16_28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A126</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="txt" href="#E144:16"><span class="lab"><font color="Green" title="E116">A123</font></span></a>, <a class="ref" href="finseq_6.html#T42" title="FINSEQ_6:th.42">FINSEQ_6:42</a></span>;<br/>
<a NAME="E4:16_28"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_28"><span class="lab"><font color="Green" title="E118">A125</font></span></a>, <a class="ref" href="revrot_1.html#T3" title="REVROT_1:th.3">REVROT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_28"/>
<font color="Maroon" title="c28">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> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:16"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="txt" href="#E137:16"><span class="lab"><font color="Green" title="E111">A114</font></span></a>, <a class="txt" href="#E3:16_28"><span class="lab"><font color="Green" title="E119">A126</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
<a NAME="E146:16"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_29"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">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> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span> or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_29"/><span class="lab"><font color="Green" title="E118">A127</font></span>: 
<font color="Maroon" title="c28">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> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:16_29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A128</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:16_29"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_29"><span class="lab"><font color="Green" title="E118">A127</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="E4:16_29"/><span class="kw">then </span>
<font color="Maroon" title="c28">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> <font color="Maroon" title="c26">pion1</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:16"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E2:16_29"><span class="lab"><font color="Green" title="E119">A128</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_29"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E55:16"><span class="lab"><font color="Green" title="E54">A42</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="txt" href="#E144:16"><span class="lab"><font color="Green" title="E116">A123</font></span></a>, <a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</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>
<a NAME="E147:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A129</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</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> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">pion1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E145:16"><span class="lab"><font color="Green" title="E117">A124</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="lab"><font color="Green" title="E119">A130</font></span>: <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></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> <font color="Maroon" title="c21">do</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span></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> <font color="Maroon" title="c21">do</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="topreal8.html#T35" title="TOPREAL8:th.35">TOPREAL8:35</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:16"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="txt" href="#E98:16"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="txt" href="#E144:16"><span class="lab"><font color="Green" title="E116">A123</font></span></a>, <a class="txt" href="#E147:16"><span class="lab"><font color="Green" title="E118">A129</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T53" title="GRAPH_2:th.53">GRAPH_2:53</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>
;<br/>
<a NAME="E149:16"/>
<font color="Maroon" title="c21">do</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">do</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:16"><span class="lab"><font color="Green" title="E68">A60</font></span></a>, <a class="ref" href="graph_2.html#T53" title="GRAPH_2:th.53">GRAPH_2:53</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">godo</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c21">do</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:16"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="txt" href="#E111:16"><span class="lab"><font color="Green" title="E97">A96</font></span></a>, <a class="txt" href="#E112:16"><span class="lab"><font color="Green" title="E98">A97</font></span></a>, <a class="txt" href="#E119:16"><span class="lab"><font color="Green" title="E101">A102</font></span></a>, <a class="txt" href="#E124:16"><span class="lab"><font color="Green" title="E103">A104</font></span></a>, <a class="txt" href="#E135:16"><span class="lab"><font color="Green" title="E109">A112</font></span></a>, <a class="txt" href="#E136:16"><span class="lab"><font color="Green" title="E110">A113</font></span></a>, <a class="txt" href="#E141:16"><span class="lab"><font color="Green" title="E114">A121</font></span></a>, <a class="txt" href="#E143:16"><span class="lab"><font color="Green" title="E115">A122</font></span></a>, <a class="txt" href="#E148:16"><span class="lab"><font color="Green" title="E119">A130</font></span></a>, <a class="ref" href="jordan8.html#T4" title="JORDAN8:th.4">JORDAN8:4</a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a>, <a class="ref" href="topreal8.html#T11" title="TOPREAL8:th.11">TOPREAL8:11</a>, <a class="ref" href="topreal8.html#T33" title="TOPREAL8:th.33">TOPREAL8:33</a>, <a class="ref" href="topreal8.html#T34" title="TOPREAL8:th.34">TOPREAL8:34</a></span>;<br/>
<a NAME="E151:16"/><span class="lab"><font color="Green" title="E120">A131</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font>, <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></span>;<br/>
<a NAME="E152:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A132</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T10" title="JORDAN6:th.10">JORDAN6:10</a></span>;<br/>
<a NAME="E153:16"/><span class="lab"><font color="Green" title="E122">A133</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E151:16"><span class="lab"><font color="Green" title="E120">A131</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<a NAME="E154:16"/><span class="lab"><font color="Green" title="E123">A134</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E151:16"><span class="lab"><font color="Green" title="E120">A131</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">ff</font> =  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>);<br/>
<a NAME="E155:16"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T43" title="SPRECT_2:th.43">SPRECT_2:43</a></span>;<br/>
<a NAME="E156:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A135</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T92" title="FINSEQ_6:th.92">FINSEQ_6:92</a></span>;<br/>
<a NAME="E157:16"/><span class="lab"><font color="Green" title="E125">A136</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T33" title="REVROT_1:th.33">REVROT_1:33</a></span>;<br/>
<a NAME="E158:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">W-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:16"><span class="lab"><font color="Green" title="E124">A135</font></span></a>, <a class="ref" href="sprect_5.html#T22" title="SPRECT_5:th.22">SPRECT_5:22</a></span>;<br/>
<a NAME="E159:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:16"><span class="lab"><font color="Green" title="E124">A135</font></span></a>, <a class="txt" href="#E157:16"><span class="lab"><font color="Green" title="E125">A136</font></span></a>, <a class="ref" href="sprect_5.html#T23" title="SPRECT_5:th.23">SPRECT_5:23</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E160:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:16"><span class="lab"><font color="Green" title="E124">A135</font></span></a>, <a class="txt" href="#E157:16"><span class="lab"><font color="Green" title="E125">A136</font></span></a>, <a class="ref" href="sprect_5.html#T24" title="SPRECT_5:th.24">SPRECT_5:24</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E161:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A137</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E156:16"><span class="lab"><font color="Green" title="E124">A135</font></span></a>, <a class="txt" href="#E157:16"><span class="lab"><font color="Green" title="E125">A136</font></span></a>, <a class="ref" href="sprect_5.html#T25" title="SPRECT_5:th.25">SPRECT_5:25</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E127">A138</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_31"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_31"/><span class="lab"><font color="Green" title="E128">A139</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <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="E2:16_31"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E3:16_31"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</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="#E1:16_31"><span class="lab"><font color="Green" title="E128">A139</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:16_31"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="finseq_5.html#T38" title="FINSEQ_5:th.38">FINSEQ_5:38</a></span>;<br/><span class="kw">hence </span><a NAME="E5:16_31"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="ref" href="jordan1f.html#T5" title="JORDAN1F:th.5">JORDAN1F: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>
<a NAME="E163:16"/><span class="lab"><font color="Green" title="E128">A140</font></span>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>) <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="c2">C</font>,<font color="Maroon" title="c1">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/>
<a NAME="E164:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A141</font></span>: 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>) <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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T34" title="REVROT_1:th.34">REVROT_1:34</a></span>;<br/>
<a NAME="E165:16"/><span class="lab"><font color="Green" title="E130">A142</font></span>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c28">godo</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c28">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:16"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="txt" href="#E112:16"><span class="lab"><font color="Green" title="E98">A97</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/>
<span class="lab"><font color="Green" title="E131">A143</font></span>:  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:16"><span class="lab"><font color="Green" title="E97">A96</font></span></a>, <a class="ref" href="topreal8.html#T35" title="TOPREAL8:th.35">TOPREAL8:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E109:16"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="ref" href="topreal8.html#T35" title="TOPREAL8:th.35">TOPREAL8:35</a></span>
;<br/>
<a NAME="E167:16"/><span class="lab"><font color="Green" title="E132">A144</font></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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#T13" title="JORDAN1E:th.13">JORDAN1E:13</a></span>;<br/>
<a NAME="E168:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A145</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E169:16"/><span class="lab"><font color="Green" title="E134">A146</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E167:16"><span class="lab"><font color="Green" title="E132">A144</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E170:16"/><span class="lab"><font color="Green" title="E135">A147</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E168:16"><span class="lab"><font color="Green" title="E133">A145</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E171:16"/><span class="lab"><font color="Green" title="E136">A148</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:16"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E169:16"><span class="lab"><font color="Green" title="E134">A146</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E172:16"/><span class="lab"><font color="Green" title="E137">A149</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_1.html#T13" title="SPRECT_1:th.13">SPRECT_1:13</a></span>;<br/>
<a NAME="E173:16"/><span class="lab"><font color="Green" title="E138">A150</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E139">A151</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_33"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_33"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font>
 ; <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="E2:16_33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A152</font></span>: 
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</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> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span> or  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E166:16"><span class="lab"><font color="Green" title="E131">A143</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_33_1"><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="E1:16_33_1"/>
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font> or  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font> or  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_33"><span class="lab"><font color="Green" title="E140">A152</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_33_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_33_1_1"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 ; <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="E2:16_33_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E170:16"><span class="lab"><font color="Green" title="E135">A147</font></span></a>, <a class="txt" href="#E172:16"><span class="lab"><font color="Green" title="E137">A149</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_33_1_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="16_33_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_33_1_2"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 ; <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"><span class="kw">hence </span><a NAME="E2:16_33_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E153:16"><span class="lab"><font color="Green" title="E122">A133</font></span></a>, <a class="txt" href="#E173:16"><span class="lab"><font color="Green" title="E138">A150</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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="16_33_1_3"><span class="kw">suppose </span></a><a NAME="E1:16_33_1_3"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 ; <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="E2:16_33_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E171:16"><span class="lab"><font color="Green" title="E136">A148</font></span></a>, <a class="txt" href="#E172:16"><span class="lab"><font color="Green" title="E137">A149</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_33_1_3"/>
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><span class="kw">end;</span></div>
 <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1) = 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E102:16"><span class="lab"><font color="Green" title="E89">A88</font></span></a>, <a class="ref" href="jordan1h.html#T23" title="JORDAN1H:th.23">JORDAN1H:23</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T28" title="REVROT_1:th.28">REVROT_1:28</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T44" title="JORDAN1H:th.44">JORDAN1H:44</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E161:16"><span class="lab"><font color="Green" title="E126">A137</font></span></a>, <a class="txt" href="#E164:16"><span class="lab"><font color="Green" title="E129">A141</font></span></a>, <a class="ref" href="jordan1j.html#T53" title="JORDAN1J:th.53">JORDAN1J:53</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#D1" title="JORDAN1E:def.1">JORDAN1E:def 1</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> (<span class="p2">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>)</span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:16"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="txt" href="#E108:16"><span class="lab"><font color="Green" title="E94">A93</font></span></a>, <a class="txt" href="#E162:16"><span class="lab"><font color="Green" title="E127">A138</font></span></a>, <a class="ref" href="jordan1j.html#T52" title="JORDAN1J:th.52">JORDAN1J:52</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:16"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E110:16"><span class="lab"><font color="Green" title="E96">A95</font></span></a>, <a class="ref" href="jordan1j.html#T51" title="JORDAN1J:th.51">JORDAN1J:51</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c28">godo</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E105:16"><span class="lab"><font color="Green" title="E91">A90</font></span></a>, <a class="txt" href="#E112:16"><span class="lab"><font color="Green" title="E98">A97</font></span></a>, <a class="ref" href="jordan1j.html#T51" title="JORDAN1J:th.51">JORDAN1J:51</a></span>
;<br/>
<a NAME="E176:16"/><span class="kw">then </span>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c28">godo</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1i.html#T6" title="JORDAN1I:th.6">JORDAN1I:6</a></span>;<br/>
<a NAME="E177:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A153</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c28">godo</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E174:16"><span class="lab"><font color="Green" title="E139">A151</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E141">A154</font></span>: <font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#T53" title="GRAPH_2:th.53">GRAPH_2:53</a></span>
<br/>.= 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:16"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="ref" href="graph_2.html#T53" title="GRAPH_2:th.53">GRAPH_2:53</a></span>
;<br/>
<a NAME="E179:16"/><span class="lab"><font color="Green" title="E142">A155</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E143">A156</font></span>: <font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">go</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:16"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="ref" href="graph_2.html#T57" title="GRAPH_2:th.57">GRAPH_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E86:16"><span class="lab"><font color="Green" title="E77">A76</font></span></a>, <a class="ref" href="graph_2.html#T57" title="GRAPH_2:th.57">GRAPH_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E179:16"><span class="lab"><font color="Green" title="E142">A155</font></span></a>, <a class="ref" href="graph_2.html#T57" title="GRAPH_2:th.57">GRAPH_2:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1e.html#T11" title="JORDAN1E:th.11">JORDAN1E:11</a></span>
;<br/>
<a NAME="E181:16"/><span class="lab"><font color="Green" title="E144">A157</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></span>;<br/>
<a NAME="E182:16"/>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:16"><span class="lab"><font color="Green" title="E69">A61</font></span></a>, <a class="txt" href="#E87:16"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E183:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A158</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E170:16"><span class="lab"><font color="Green" title="E135">A147</font></span></a>, <a class="txt" href="#E171:16"><span class="lab"><font color="Green" title="E136">A148</font></span></a>, <a class="txt" href="#E181:16"><span class="lab"><font color="Green" title="E144">A157</font></span></a>, <a class="ref" href="jordan1j.html#T21" title="JORDAN1J:th.21">JORDAN1J:21</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E184:16"/><span class="lab"><font color="Green" title="E146">A159</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E185:16"/><span class="lab"><font color="Green" title="E147">A160</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/>
<a NAME="E186:16"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="#E90:16"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="ref" href="sprect_1.html#T54" title="SPRECT_1:th.54">SPRECT_1:54</a></span>;<br/>
<a NAME="E187:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A161</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="ref" href="sppol_2.html#T21" title="SPPOL_2:th.21">SPPOL_2:21</a></span>;<br/>
<a NAME="E188:16"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E168:16"><span class="lab"><font color="Green" title="E133">A145</font></span></a>, <a class="ref" href="pscomp_1.html#T24" title="PSCOMP_1:th.24">PSCOMP_1:24</a></span>;<br/>
<a NAME="E189:16"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:16"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E190:16"/><span class="kw">then </span>
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E181:16"><span class="lab"><font color="Green" title="E144">A157</font></span></a>, <a class="txt" href="#E183:16"><span class="lab"><font color="Green" title="E145">A158</font></span></a>, <a class="txt" href="#E184:16"><span class="lab"><font color="Green" title="E146">A159</font></span></a>, <a class="txt" href="#E185:16"><span class="lab"><font color="Green" title="E147">A160</font></span></a>, <a class="txt" href="#E187:16"><span class="lab"><font color="Green" title="E148">A161</font></span></a>, <a class="ref" href="jordan1j.html#T33" title="JORDAN1J:th.33">JORDAN1J:33</a></span>;<br/>
<a NAME="E191:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A162</font></span>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E166:16"><span class="lab"><font color="Green" title="E131">A143</font></span></a>, <a class="txt" href="#E183:16"><span class="lab"><font color="Green" title="E145">A158</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<a NAME="E192:16"/><span class="lab"><font color="Green" title="E150">A163</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">godo</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:16"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="txt" href="#E106:16"><span class="lab"><font color="Green" title="E92">A91</font></span></a>, <a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E193:16"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:16"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E194:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A164</font></span>: 
<font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" title="PARTFUN2:th.2">PARTFUN2:2</a></span>;<br/>
<a NAME="E195:16"/>
<font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E180:16"><span class="lab"><font color="Green" title="E143">A156</font></span></a>, <a class="ref" href="jordan1i.html#T25" title="JORDAN1I:th.25">JORDAN1I:25</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></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="#E191:16"><span class="lab"><font color="Green" title="E149">A162</font></span></a>, <a class="ref" href="pscomp_1.html#T31" title="PSCOMP_1:th.31">PSCOMP_1:31</a></span>
<br/>.= 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/>
<a NAME="E197:16"/><span class="kw">then </span>
<font color="Maroon" title="c28">godo</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E192:16"><span class="lab"><font color="Green" title="E150">A163</font></span></a>, <a class="txt" href="#E194:16"><span class="lab"><font color="Green" title="E151">A164</font></span></a>, <a class="ref" href="sprect_2.html#T12" title="SPRECT_2:th.12">SPRECT_2:12</a></span>;<br/>
<a NAME="E198:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<font color="Maroon" title="c28">godo</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">W-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c28">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E178:16"><span class="lab"><font color="Green" title="E141">A154</font></span></a>, <a class="txt" href="#E191:16"><span class="lab"><font color="Green" title="E149">A162</font></span></a>, <a class="ref" href="finseq_6.html#T89" title="FINSEQ_6:th.89">FINSEQ_6:89</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">godo</font> = <font color="Maroon" title="c28">godo</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>   <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1i.html#T25" title="JORDAN1I:th.25">JORDAN1I:25</a></span>;<br/>
<a NAME="E200:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E152">A165</font></span>: <span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1f.html#T7" title="JORDAN1F:th.7">JORDAN1F:7</a></span>
;<br/>
<a NAME="E202:16"/><span class="lab"><font color="Green" title="E153">A166</font></span>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_39"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:16_39"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_39"/><span class="lab"><font color="Green" title="E154">A167</font></span>: 
<font color="Maroon" title="c31">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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:16_39"/><span class="lab"><font color="Green" title="E155">A168</font></span>: 
<font color="Maroon" title="c31">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">p</font> = <font color="Maroon" title="c31">p</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_39"><span class="lab"><font color="Green" title="E154">A167</font></span></a></span>;<br/>
<a NAME="E6:16_39"/>
<font color="Maroon" title="c32">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="jordan1e.html#K1" title="JORDAN1E:func.1">Upper_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E4:16_39"><span class="lab"><font color="Green" title="E155">A168</font></span></a></span>;<br/>
<a NAME="E7:16_39"/><span class="kw">then </span>
<font color="Maroon" title="c32">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> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span></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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E168:16"><span class="lab"><font color="Green" title="E133">A145</font></span></a>, <a class="txt" href="#E3:16_39"><span class="lab"><font color="Green" title="E154">A167</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="E8:16_39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A169</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1a.html#T83" title="JORDAN1A:th.83">JORDAN1A:83</a>, <a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></span>;<br/>
<a NAME="E9:16_39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A170</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E4:16_39"><span class="lab"><font color="Green" title="E155">A168</font></span></a>, <a class="ref" href="jordan1j.html#T46" title="JORDAN1J:th.46">JORDAN1J:46</a></span>;<br/>
<a NAME="E10:16_39"/><span class="kw">then </span>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E201:16"><span class="lab"><font color="Green" title="E152">A165</font></span></a>, <a class="txt" href="#E4:16_39"><span class="lab"><font color="Green" title="E155">A168</font></span></a>, <a class="ref" href="jordan1j.html#T43" title="JORDAN1J:th.43">JORDAN1J:43</a></span>;<br/>
<a NAME="E11:16_39"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c5">k</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="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E8:16_39"><span class="lab"><font color="Green" title="E156">A169</font></span></a>, <a class="txt" href="#E9:16_39"><span class="lab"><font color="Green" title="E157">A170</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_39"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</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="hs2(this)" href="javascript:()" title="16_40"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_40"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font>
 ; <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="E2:16_40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A171</font></span>: 
(  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span> or  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E166:16"><span class="lab"><font color="Green" title="E131">A143</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_40_1"><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="E1:16_40_1"/>
(  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font> or  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font> or  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_40"><span class="lab"><font color="Green" title="E154">A171</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_40_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_40_1_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 ; <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"><span class="kw">hence </span><a NAME="E2:16_40_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E202:16"><span class="lab"><font color="Green" title="E153">A166</font></span></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="16_40_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_40_1_2"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 ; <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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_40_1_2"/><span class="lab"><font color="Green" title="E155">A172</font></span>: 
<font color="Maroon" title="c31">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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:16_40_1_2"/><span class="lab"><font color="Green" title="E156">A173</font></span>: 
<font color="Maroon" title="c31">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c32">p</font> = <font color="Maroon" title="c31">p</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_40_1_2"><span class="lab"><font color="Green" title="E155">A172</font></span></a></span>;<br/><a NAME="E6:16_40_1_2"/><span class="lab"><font color="Green" title="E157">A174</font></span>: 
<font color="Maroon" title="c32">p</font> <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</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="#E90:16"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E173:16"><span class="lab"><font color="Green" title="E138">A150</font></span></a>, <a class="txt" href="#E4:16_40_1_2"><span class="lab"><font color="Green" title="E156">A173</font></span></a>, <a class="ref" href="goboard7.html#T5" title="GOBOARD7:th.5">GOBOARD7:5</a></span>;<br/><a NAME="E7:16_40_1_2"/>
<font color="Maroon" title="c3">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:16_40_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E9:16_40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A175</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E10:16_40_1_2"/>
<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="c2">C</font>,<font color="Maroon" title="c1">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="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>,<font color="Maroon" title="c1">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="E11:16_40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1)</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="#E1:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E6:16_40_1_2"><span class="lab"><font color="Green" title="E157">A174</font></span></a>, <a class="txt" href="#E9:16_40_1_2"><span class="lab"><font color="Green" title="E158">A175</font></span></a>, <a class="ref" href="jordan1a.html#T18" title="JORDAN1A:th.18">JORDAN1A:18</a></span>;<br/><a NAME="E12:16_40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="jordan8.html#T12" title="JORDAN8:th.12">JORDAN8:12</a></span>;<br/><a NAME="E13:16_40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A176</font></span>: 
<font color="Maroon" title="c32">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"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">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="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>;<br/><a NAME="E14:16_40_1_2"/>
<font color="Maroon" title="c32">p</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"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">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="#E3:16_40_1_2"><span class="lab"><font color="Green" title="E155">A172</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>;<br/><a NAME="E15:16_40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E160">A177</font></span>: 
<font color="Maroon" title="c32">p</font> <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"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">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="#E13:16_40_1_2"><span class="lab"><font color="Green" title="E159">A176</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:16_40_1_2"/>
<font color="Maroon" title="c32">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</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="#E3:16_40_1_2"><span class="lab"><font color="Green" title="E155">A172</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>;<br/><a NAME="E17:16_40_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16_40_1_2"><span class="lab"><font color="Green" title="E160">A177</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E18:16_40_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E154:16"><span class="lab"><font color="Green" title="E123">A134</font></span></a>, <a class="txt" href="#E173:16"><span class="lab"><font color="Green" title="E138">A150</font></span></a>, <a class="txt" href="#E4:16_40_1_2"><span class="lab"><font color="Green" title="E156">A173</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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="16_40_1_3"><span class="kw">suppose </span></a><a NAME="E1:16_40_1_3"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 ; <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"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_40_1_3"/><span class="lab"><font color="Green" title="E155">A178</font></span>: 
<font color="Maroon" title="c31">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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:16_40_1_3"/><span class="lab"><font color="Green" title="E156">A179</font></span>: 
<font color="Maroon" title="c31">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c32">p</font> = <font color="Maroon" title="c31">p</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_40_1_3"><span class="lab"><font color="Green" title="E155">A178</font></span></a></span>;<br/><a NAME="E6:16_40_1_3"/>
<font color="Maroon" title="c32">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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:16"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E4:16_40_1_3"><span class="lab"><font color="Green" title="E156">A179</font></span></a></span>;<br/><a NAME="E7:16_40_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c32">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> <span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span></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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E169:16"><span class="lab"><font color="Green" title="E134">A146</font></span></a>, <a class="txt" href="#E3:16_40_1_3"><span class="lab"><font color="Green" title="E155">A178</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="E8:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A180</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1a.html#T83" title="JORDAN1A:th.83">JORDAN1A:83</a>, <a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></span>;<br/><a NAME="E9:16_40_1_3"/><span class="lab"><font color="Green" title="E158">A181</font></span>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">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="#E3:16_40_1_3"><span class="lab"><font color="Green" title="E155">A178</font></span></a>, <a class="ref" href="topreal1.html#D11" title="TOPREAL1:def.11">TOPREAL1:def 11</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c33">RC</font> =  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>);<br/><a NAME="E10:16_40_1_3"/><span class="lab"><font color="Green" title="E159">A182</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1i.html#T7" title="JORDAN1I:th.7">JORDAN1I:7</a></span>;<br/><a NAME="E11:16_40_1_3"/><span class="lab"><font color="Green" title="E160">A183</font></span>: 
1 <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:16_40_1_3"/>
 <a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1g.html#T18" title="JORDAN1G:th.18">JORDAN1G:18</a></span>;<br/><a NAME="E13:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A184</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_40_1_3"><span class="lab"><font color="Green" title="E160">A183</font></span></a>, <a class="ref" href="sppol_2.html#T9" title="SPPOL_2:th.9">SPPOL_2:9</a></span>;<br/><a NAME="E14:16_40_1_3"/><span class="lab"><font color="Green" title="E162">A185</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T33" title="REVROT_1:th.33">REVROT_1:33</a></span>;<br/><a NAME="E15:16_40_1_3"/><span class="lab"><font color="Green" title="E163">A186</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T14" title="REVROT_1:th.14">REVROT_1:14</a></span>;<br/><span class="lab"><font color="Green" title="E164">A187</font></span>:  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> = 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T28" title="REVROT_1:th.28">REVROT_1:28</a></span>
<br/>.= 
 <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T44" title="JORDAN1H:th.44">JORDAN1H:44</a></span>
;<br/><a NAME="E17:16_40_1_3"/><span class="lab"><font color="Green" title="E165">A188</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T46" title="SPRECT_2:th.46">SPRECT_2:46</a></span>;<br/><a NAME="E18:16_40_1_3"/><span class="lab"><font color="Green" title="E166">A189</font></span>: 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>) <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="c2">C</font>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E163:16"><span class="lab"><font color="Green" title="E128">A140</font></span></a>, <a class="ref" href="revrot_1.html#T34" title="REVROT_1:th.34">REVROT_1:34</a></span>;<br/><a NAME="E19:16_40_1_3"/><span class="lab"><font color="Green" title="E167">A190</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_40_1_3"><span class="lab"><font color="Green" title="E162">A185</font></span></a>, <a class="txt" href="#E17:16_40_1_3"><span class="lab"><font color="Green" title="E165">A188</font></span></a>, <a class="ref" href="finseq_6.html#T92" title="FINSEQ_6:th.92">FINSEQ_6:92</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c34">ii</font>, <font color="Maroon" title="c35">jj</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E21:16_40_1_3"/><span class="lab"><font color="Green" title="E168">A191</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">ii</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:16_40_1_3"/><span class="lab"><font color="Green" title="E169">A192</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">ii</font>,<font color="Maroon" title="c35">jj</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E23:16_40_1_3"/><span class="lab"><font color="Green" title="E170">A193</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c34">ii</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c35">jj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E24:16_40_1_3"/><span class="lab"><font color="Green" title="E171">A194</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c34">ii</font>,<font color="Maroon" title="c35">jj</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:16"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E14:16_40_1_3"><span class="lab"><font color="Green" title="E162">A185</font></span></a>, <a class="txt" href="#E15:16_40_1_3"><span class="lab"><font color="Green" title="E163">A186</font></span></a>, <a class="txt" href="#E17:16_40_1_3"><span class="lab"><font color="Green" title="E165">A188</font></span></a>, <a class="txt" href="#E18:16_40_1_3"><span class="lab"><font color="Green" title="E166">A189</font></span></a>, <a class="ref" href="finseq_6.html#T92" title="FINSEQ_6:th.92">FINSEQ_6:92</a>, <a class="ref" href="jordan1i.html#T23" title="JORDAN1I:th.23">JORDAN1I:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c36">jj2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E26:16_40_1_3"/><span class="lab"><font color="Green" title="E172">A195</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">jj2</font>
 <span class="kw">and </span><br/><a NAME="E27:16_40_1_3"/><span class="lab"><font color="Green" title="E173">A196</font></span>: 
<font color="Maroon" title="c36">jj2</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E28:16_40_1_3"/><span class="lab"><font color="Green" title="E174">A197</font></span>: 
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c36">jj2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1d.html#T25" title="JORDAN1D:th.25">JORDAN1D:25</a></span>;<br/><a NAME="E29:16_40_1_3"/><span class="lab"><font color="Green" title="E175">A198</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T10" title="JORDAN8:th.10">JORDAN8:10</a></span>;<br/><a NAME="E30:16_40_1_3"/><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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:16_40_1_3"/><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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c36">jj2</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16_40_1_3"><span class="lab"><font color="Green" title="E172">A195</font></span></a>, <a class="txt" href="#E27:16_40_1_3"><span class="lab"><font color="Green" title="E173">A196</font></span></a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>;<br/><a NAME="E32:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A199</font></span>: 
<font color="Maroon" title="c34">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_40_1_3"><span class="lab"><font color="Green" title="E162">A185</font></span></a>, <a class="txt" href="#E19:16_40_1_3"><span class="lab"><font color="Green" title="E167">A190</font></span></a>, <a class="txt" href="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="txt" href="#E23:16_40_1_3"><span class="lab"><font color="Green" title="E170">A193</font></span></a>, <a class="txt" href="#E28:16_40_1_3"><span class="lab"><font color="Green" title="E174">A197</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E33:16_40_1_3"/><span class="lab"><font color="Green" title="E177">A200</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">ii</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E34:16_40_1_3"/><span class="lab"><font color="Green" title="E178">A201</font></span>: 
<font color="Maroon" title="c34">ii</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>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E35:16_40_1_3"/><span class="lab"><font color="Green" title="E179">A202</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">jj</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="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E36:16_40_1_3"/><span class="lab"><font color="Green" title="E180">A203</font></span>: 
<font color="Maroon" title="c35">jj</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="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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E37:16_40_1_3"/><span class="lab"><font color="Green" title="E181">A204</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">ii</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_40_1_3"><span class="lab"><font color="Green" title="E169">A192</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E38:16_40_1_3"/><span class="lab"><font color="Green" title="E182">A205</font></span>: 
<font color="Maroon" title="c34">ii</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>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_40_1_3"><span class="lab"><font color="Green" title="E169">A192</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E39:16_40_1_3"/><span class="lab"><font color="Green" title="E183">A206</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">jj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_40_1_3"><span class="lab"><font color="Green" title="E169">A192</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E40:16_40_1_3"/><span class="lab"><font color="Green" title="E184">A207</font></span>: 
<font color="Maroon" title="c35">jj</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_40_1_3"><span class="lab"><font color="Green" title="E169">A192</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E41:16_40_1_3"/><span class="lab"><font color="Green" title="E185">A208</font></span>: 
<font color="Maroon" title="c34">ii</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c34">ii</font>
 ;<br/><a NAME="E42:16_40_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c35">jj</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">jj</font>
 ;<br/><a NAME="E43:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E186">A209</font></span>: 
 <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</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>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c35">jj</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:16"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E15:16_40_1_3"><span class="lab"><font color="Green" title="E163">A186</font></span></a>, <a class="txt" href="#E16:16_40_1_3"><span class="lab"><font color="Green" title="E164">A187</font></span></a>, <a class="txt" href="#E21:16_40_1_3"><span class="lab"><font color="Green" title="E168">A191</font></span></a>, <a class="txt" href="#E22:16_40_1_3"><span class="lab"><font color="Green" title="E169">A192</font></span></a>, <a class="txt" href="#E23:16_40_1_3"><span class="lab"><font color="Green" title="E170">A193</font></span></a>, <a class="txt" href="#E24:16_40_1_3"><span class="lab"><font color="Green" title="E171">A194</font></span></a>, <a class="txt" href="#E41:16_40_1_3"><span class="lab"><font color="Green" title="E185">A208</font></span></a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></span>;<br/><a NAME="E44:16_40_1_3"/><span class="lab"><font color="Green" title="E187">A210</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">ii</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:16_40_1_3"><span class="lab"><font color="Green" title="E177">A200</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E45:16_40_1_3"/>
<font color="Maroon" title="c34">ii</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:16_40_1_3"><span class="lab"><font color="Green" title="E175">A198</font></span></a>, <a class="txt" href="#E32:16_40_1_3"><span class="lab"><font color="Green" title="E176">A199</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E46:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E188">A211</font></span>: 
<font color="Maroon" title="c34">ii</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E47:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E189">A212</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E48:16_40_1_3"/><span class="lab"><font color="Green" title="E190">A213</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c35">jj</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">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="#E9:16_40_1_3"><span class="lab"><font color="Green" title="E158">A181</font></span></a>, <a class="txt" href="#E10:16_40_1_3"><span class="lab"><font color="Green" title="E159">A182</font></span></a>, <a class="txt" href="#E34:16_40_1_3"><span class="lab"><font color="Green" title="E178">A201</font></span></a>, <a class="txt" href="#E36:16_40_1_3"><span class="lab"><font color="Green" title="E180">A203</font></span></a>, <a class="txt" href="#E39:16_40_1_3"><span class="lab"><font color="Green" title="E183">A206</font></span></a>, <a class="txt" href="#E43:16_40_1_3"><span class="lab"><font color="Green" title="E186">A209</font></span></a>, <a class="txt" href="#E44:16_40_1_3"><span class="lab"><font color="Green" title="E187">A210</font></span></a>, <a class="txt" href="#E46:16_40_1_3"><span class="lab"><font color="Green" title="E188">A211</font></span></a>, <a class="ref" href="jordan9.html#T17" title="JORDAN9:th.17">JORDAN9:17</a></span>;<br/><a NAME="E49:16_40_1_3"/><span class="lab"><font color="Green" title="E191">A214</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</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="#E9:16_40_1_3"><span class="lab"><font color="Green" title="E158">A181</font></span></a>, <a class="txt" href="#E10:16_40_1_3"><span class="lab"><font color="Green" title="E159">A182</font></span></a>, <a class="txt" href="#E34:16_40_1_3"><span class="lab"><font color="Green" title="E178">A201</font></span></a>, <a class="txt" href="#E36:16_40_1_3"><span class="lab"><font color="Green" title="E180">A203</font></span></a>, <a class="txt" href="#E39:16_40_1_3"><span class="lab"><font color="Green" title="E183">A206</font></span></a>, <a class="txt" href="#E43:16_40_1_3"><span class="lab"><font color="Green" title="E186">A209</font></span></a>, <a class="txt" href="#E44:16_40_1_3"><span class="lab"><font color="Green" title="E187">A210</font></span></a>, <a class="txt" href="#E46:16_40_1_3"><span class="lab"><font color="Green" title="E188">A211</font></span></a>, <a class="ref" href="jordan9.html#T17" title="JORDAN9:th.17">JORDAN9:17</a></span>;<br/><a NAME="E50:16_40_1_3"/><span class="lab"><font color="Green" title="E192">A215</font></span>: 
<font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16_40_1_3"><span class="lab"><font color="Green" title="E178">A201</font></span></a>, <a class="txt" href="#E44:16_40_1_3"><span class="lab"><font color="Green" title="E187">A210</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E193">A216</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c35">jj</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c35">jj</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="#E39:16_40_1_3"><span class="lab"><font color="Green" title="E183">A206</font></span></a>, <a class="txt" href="#E40:16_40_1_3"><span class="lab"><font color="Green" title="E184">A207</font></span></a>, <a class="txt" href="#E47:16_40_1_3"><span class="lab"><font color="Green" title="E189">A212</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c34">ii</font>,<font color="Maroon" title="c35">jj</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="#E37:16_40_1_3"><span class="lab"><font color="Green" title="E181">A204</font></span></a>, <a class="txt" href="#E38:16_40_1_3"><span class="lab"><font color="Green" title="E182">A205</font></span></a>, <a class="txt" href="#E39:16_40_1_3"><span class="lab"><font color="Green" title="E183">A206</font></span></a>, <a class="txt" href="#E40:16_40_1_3"><span class="lab"><font color="Green" title="E184">A207</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>
;<br/><span class="lab"><font color="Green" title="E194">A217</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c34">ii</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</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="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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</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="#E35:16_40_1_3"><span class="lab"><font color="Green" title="E179">A202</font></span></a>, <a class="txt" href="#E36:16_40_1_3"><span class="lab"><font color="Green" title="E180">A203</font></span></a>, <a class="txt" href="#E47:16_40_1_3"><span class="lab"><font color="Green" title="E189">A212</font></span></a>, <a class="txt" href="#E50:16_40_1_3"><span class="lab"><font color="Green" title="E192">A215</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c34">ii</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</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="#E33:16_40_1_3"><span class="lab"><font color="Green" title="E177">A200</font></span></a>, <a class="txt" href="#E34:16_40_1_3"><span class="lab"><font color="Green" title="E178">A201</font></span></a>, <a class="txt" href="#E35:16_40_1_3"><span class="lab"><font color="Green" title="E179">A202</font></span></a>, <a class="txt" href="#E36:16_40_1_3"><span class="lab"><font color="Green" title="E180">A203</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>
;<br/><a NAME="E53:16_40_1_3"/><span class="lab"><font color="Green" title="E195">A218</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c35">jj</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E39:16_40_1_3"><span class="lab"><font color="Green" title="E183">A206</font></span></a>, <a class="txt" href="#E40:16_40_1_3"><span class="lab"><font color="Green" title="E184">A207</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/><a NAME="E54:16_40_1_3"/>
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c35">jj</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="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E35:16_40_1_3"><span class="lab"><font color="Green" title="E179">A202</font></span></a>, <a class="txt" href="#E36:16_40_1_3"><span class="lab"><font color="Green" title="E180">A203</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/><a NAME="E55:16_40_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> (<span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <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="#E8:16_40_1_3"><span class="lab"><font color="Green" title="E157">A180</font></span></a>, <a class="txt" href="#E23:16_40_1_3"><span class="lab"><font color="Green" title="E170">A193</font></span></a>, <a class="txt" href="#E24:16_40_1_3"><span class="lab"><font color="Green" title="E171">A194</font></span></a>, <a class="txt" href="#E32:16_40_1_3"><span class="lab"><font color="Green" title="E176">A199</font></span></a>, <a class="txt" href="#E48:16_40_1_3"><span class="lab"><font color="Green" title="E190">A213</font></span></a>, <a class="txt" href="#E49:16_40_1_3"><span class="lab"><font color="Green" title="E191">A214</font></span></a>, <a class="txt" href="#E51:16_40_1_3"><span class="lab"><font color="Green" title="E193">A216</font></span></a>, <a class="txt" href="#E52:16_40_1_3"><span class="lab"><font color="Green" title="E194">A217</font></span></a>, <a class="txt" href="#E53:16_40_1_3"><span class="lab"><font color="Green" title="E195">A218</font></span></a>, <a class="ref" href="goboard7.html#T7" title="GOBOARD7:th.7">GOBOARD7:7</a></span>;<br/><a NAME="E56:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E196">A219</font></span>: 
<font color="Maroon" title="c32">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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E101:16"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E13:16_40_1_3"><span class="lab"><font color="Green" title="E161">A184</font></span></a>, <a class="txt" href="#E15:16_40_1_3"><span class="lab"><font color="Green" title="E163">A186</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E57:16_40_1_3"/><span class="lab"><font color="Green" title="E197">A220</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">do</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_40_1_3"><span class="lab"><font color="Green" title="E156">A179</font></span></a>, <a class="ref" href="jordan3.html#T9" title="JORDAN3:th.9">JORDAN3:9</a></span>;<br/><a NAME="E58:16_40_1_3"/><span class="lab"><font color="Green" title="E198">A221</font></span>: 
<font color="Maroon" title="c21">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="jordan1j.html#T37" title="JORDAN1J:th.37">JORDAN1J:37</a></span>;<br/><a NAME="E59:16_40_1_3"/><span class="lab"><font color="Green" title="E199">A222</font></span>: 
1 <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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E60:16_40_1_3"/><span class="lab"><font color="Green" title="E200">A223</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="finseq_4.html#T21" title="FINSEQ_4:th.21">FINSEQ_4:21</a></span>;<br/><a NAME="E61:16_40_1_3"/>
<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:16"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="finseq_4.html#T19" title="FINSEQ_4:th.19">FINSEQ_4:19</a></span>;<br/><a NAME="E62:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E201">A224</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16_40_1_3"><span class="lab"><font color="Green" title="E200">A223</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E63:16_40_1_3"/><span class="lab"><font color="Green" title="E202">A225</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_40_1_3"><span class="lab"><font color="Green" title="E156">A179</font></span></a>, <a class="ref" href="jordan3.html#T8" title="JORDAN3:th.8">JORDAN3:8</a></span>;<br/><a NAME="E64:16_40_1_3"/><span class="lab"><font color="Green" title="E203">A226</font></span>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</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> <font color="Maroon" title="c21">do</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_40_1_3"><span class="lab"><font color="Green" title="E156">A179</font></span></a>, <a class="ref" href="jordan3.html#T8" title="JORDAN3:th.8">JORDAN3:8</a></span>;<br/><a NAME="E65:16_40_1_3"/><span class="lab"><font color="Green" title="E204">A227</font></span>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:16"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="jordan1j.html#T56" title="JORDAN1J:th.56">JORDAN1J:56</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c37">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E67:16_40_1_3"/><span class="lab"><font color="Green" title="E205">A228</font></span>: 
<font color="Maroon" title="c37">t</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E68:16_40_1_3"/><span class="lab"><font color="Green" title="E206">A229</font></span>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">t</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/><a NAME="E69:16_40_1_3"/><span class="lab"><font color="Green" title="E207">A230</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:16_40_1_3"><span class="lab"><font color="Green" title="E205">A228</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E70:16_40_1_3"/><span class="lab"><font color="Green" title="E208">A231</font></span>: 
<font color="Maroon" title="c37">t</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:16_40_1_3"><span class="lab"><font color="Green" title="E205">A228</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E71:16_40_1_3"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c37">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:16"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E68:16_40_1_3"><span class="lab"><font color="Green" title="E206">A229</font></span></a>, <a class="txt" href="#E69:16_40_1_3"><span class="lab"><font color="Green" title="E207">A230</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E72:16_40_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:16_40_1_3"><span class="lab"><font color="Green" title="E206">A229</font></span></a>, <a class="txt" href="#E70:16_40_1_3"><span class="lab"><font color="Green" title="E208">A231</font></span></a>, <a class="ref" href="jordan3.html#T12" title="JORDAN3:th.12">JORDAN3:12</a></span>;<br/><a NAME="E73:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E209">A232</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> (<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E68:16_40_1_3"><span class="lab"><font color="Green" title="E206">A229</font></span></a>, <a class="ref" href="jordan3.html#T26" title="JORDAN3:th.26">JORDAN3:26</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c38">tt</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/><a NAME="E74:16_40_1_3"/><span class="lab"><font color="Green" title="E210">A233</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan3.html#T8" title="JORDAN3:th.8">JORDAN3:8</a></span>;<br/><a NAME="E75:16_40_1_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="jordan3.html#T8" title="JORDAN3:th.8">JORDAN3:8</a></span>;<br/><a NAME="E76:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E211">A234</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E77:16_40_1_3"/>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:16_40_1_3"><span class="lab"><font color="Green" title="E203">A226</font></span></a>, <a class="txt" href="#E73:16_40_1_3"><span class="lab"><font color="Green" title="E209">A232</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E78:16_40_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <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> <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E79:16_40_1_3"/><span class="kw">then </span>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E80:16_40_1_3"/><span class="kw">then </span>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:16_40_1_3"><span class="lab"><font color="Green" title="E211">A234</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E81:16_40_1_3"/><span class="kw">then </span>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:16_40_1_3"><span class="lab"><font color="Green" title="E204">A227</font></span></a></span>;<br/><a NAME="E82:16_40_1_3"/><span class="kw">then </span>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E83:16_40_1_3"/><span class="kw">then </span>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>) <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E84:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E212">A235</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:16_40_1_3"><span class="lab"><font color="Green" title="E199">A222</font></span></a>, <a class="txt" href="#E62:16_40_1_3"><span class="lab"><font color="Green" title="E201">A224</font></span></a>, <a class="txt" href="#E63:16_40_1_3"><span class="lab"><font color="Green" title="E202">A225</font></span></a>, <a class="ref" href="jordan4.html#T19" title="JORDAN4:th.19">JORDAN4:19</a></span>;<br/><a NAME="E85:16_40_1_3"/><span class="lab"><font color="Green" title="E213">A236</font></span>: 
1 <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> <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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:16_40_1_3"><span class="lab"><font color="Green" title="E204">A227</font></span></a>, <a class="txt" href="#E74:16_40_1_3"><span class="lab"><font color="Green" title="E210">A233</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E86:16_40_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E63:16_40_1_3"><span class="lab"><font color="Green" title="E202">A225</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E87:16_40_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <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">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E88:16_40_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E214">A237</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E89:16_40_1_3"/><span class="lab"><font color="Green" title="E215">A238</font></span>: 
2 <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="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_40_1_3"><span class="lab"><font color="Green" title="E160">A183</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_40_1_3_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_40_1_3_4_1"><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="E1:16_40_1_3_4_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:16_40_1_3"><span class="lab"><font color="Green" title="E214">A237</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_40_1_3_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_40_1_3_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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="E2:16_40_1_3_4_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_40_1_3_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:16_40_1_3"><span class="lab"><font color="Green" title="E196">A219</font></span></a>, <a class="txt" href="#E57:16_40_1_3"><span class="lab"><font color="Green" title="E197">A220</font></span></a>, <a class="txt" href="#E58:16_40_1_3"><span class="lab"><font color="Green" title="E198">A221</font></span></a>, <a class="txt" href="#E84:16_40_1_3"><span class="lab"><font color="Green" title="E212">A235</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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="16_40_1_3_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_40_1_3_4_1_2"/><span class="lab"><font color="Green" title="E216">A239</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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="E2:16_40_1_3_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,1)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p5">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/><a NAME="E3:16_40_1_3_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:16_40_1_3"><span class="lab"><font color="Green" title="E196">A219</font></span></a>, <a class="txt" href="#E57:16_40_1_3"><span class="lab"><font color="Green" title="E197">A220</font></span></a>, <a class="txt" href="#E58:16_40_1_3"><span class="lab"><font color="Green" title="E198">A221</font></span></a>, <a class="txt" href="#E84:16_40_1_3"><span class="lab"><font color="Green" title="E212">A235</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="E4:16_40_1_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E217">A240</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:16_40_1_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E218">A241</font></span>: 
<font color="Maroon" title="c32">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:16_40_1_3"><span class="lab"><font color="Green" title="E215">A238</font></span></a>, <a class="ref" href="finseq_5.html#T41" title="FINSEQ_5:th.41">FINSEQ_5:41</a></span>;<br/><a NAME="E6:16_40_1_3_4_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_40_1_3_4_1_2"><span class="lab"><font color="Green" title="E216">A239</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E7:16_40_1_3_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> (<font color="Maroon" title="c32">p</font>,<font color="Maroon" title="c21">do</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 ;<br/><a NAME="E8:16_40_1_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E219">A242</font></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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:16_40_1_3"><span class="lab"><font color="Green" title="E202">A225</font></span></a>, <a class="txt" href="#E85:16_40_1_3"><span class="lab"><font color="Green" title="E213">A236</font></span></a>, <a class="ref" href="jordan1e.html#T6" title="JORDAN1E:th.6">JORDAN1E:6</a></span>;<br/><a NAME="E9:16_40_1_3_4_1_2"/>
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2" title="JORDAN1E:func.2">Lower_Seq</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E89:16_40_1_3"><span class="lab"><font color="Green" title="E215">A238</font></span></a>, <a class="txt" href="#E4:16_40_1_3_4_1_2"><span class="lab"><font color="Green" title="E217">A240</font></span></a>, <a class="ref" href="partfun2.html#T2" title="PARTFUN2:th.2">PARTFUN2:2</a></span>;<br/><a NAME="E10:16_40_1_3_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c32">p</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="txt" href="#E5:16_40_1_3_4_1_2"><span class="lab"><font color="Green" title="E218">A241</font></span></a>, <a class="txt" href="#E8:16_40_1_3_4_1_2"><span class="lab"><font color="Green" title="E219">A242</font></span></a>, <a class="ref" href="finseq_5.html#T9" title="FINSEQ_5:th.9">FINSEQ_5:9</a></span>;<br/><a NAME="E11:16_40_1_3_4_1_2"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_40_1_3_4_1_2"><span class="lab"><font color="Green" title="E217">A240</font></span></a>, <a class="ref" href="jordan1g.html#T32" title="JORDAN1G:th.32">JORDAN1G:32</a></span>;<br/><a NAME="E12:16_40_1_3_4_1_2"/><span class="kw">then </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>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c4">j</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="#E5:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="jordan1a.html#T71" title="JORDAN1A:th.71">JORDAN1A:71</a></span>;<br/><span class="kw">hence </span><a NAME="E13:16_40_1_3_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E80:16"><span class="lab"><font color="Green" title="E73">A67</font></span></a>, <a class="ref" href="jordan1g.html#T7" title="JORDAN1G:th.7">JORDAN1G:7</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E91:16_40_1_3"/>
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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E204:16"/><span class="kw">then </span>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">W</font> being   <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><span class="kw"> such that </span><br/><a NAME="E206:16"/><span class="lab"><font color="Green" title="E154">A243</font></span>: 
<font color="Maroon" title="c31">W</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">and </span><br/><a NAME="E207:16"/><span class="lab"><font color="Green" title="E155">A244</font></span>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c31">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#T3" title="GOBOARD9:th.3">GOBOARD9:3</a></span>;<br/>
<a NAME="E208:16"/>
 not <font color="Maroon" title="c31">W</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E207:16"><span class="lab"><font color="Green" title="E155">A244</font></span></a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a>, <a class="ref" href="jordan2c.html#T121" title="JORDAN2C:th.121">JORDAN2C:121</a></span>;<br/>
<a NAME="E209:16"/><span class="kw">then </span>
<font color="Maroon" title="c31">W</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E206:16"><span class="lab"><font color="Green" title="E154">A243</font></span></a>, <a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a></span>;<br/>
<a NAME="E210:16"/><span class="kw">then </span>
<font color="Maroon" title="c31">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T23" title="JORDAN2C:th.23">JORDAN2C:23</a></span>;<br/>
<a NAME="E211:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A245</font></span>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E207:16"><span class="lab"><font color="Green" title="E155">A244</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E212:16"/>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T38" title="TOPREAL1:th.38">TOPREAL1:38</a></span>;<br/>
<a NAME="E213:16"/><span class="kw">then </span>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E211:16"><span class="lab"><font color="Green" title="E156">A245</font></span></a></span>;<br/>
<a NAME="E214:16"/><span class="kw">then </span>
 <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c30">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T36" title="GOBRD14:th.36">GOBRD14:36</a></span>;<br/>
<a NAME="E215:16"/><span class="kw">then </span>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c30">godo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:16"><span class="lab"><font color="Green" title="E121">A132</font></span></a>, <a class="txt" href="#E153:16"><span class="lab"><font color="Green" title="E122">A133</font></span></a>, <a class="txt" href="#E154:16"><span class="lab"><font color="Green" title="E123">A134</font></span></a>, <a class="txt" href="#E165:16"><span class="lab"><font color="Green" title="E130">A142</font></span></a>, <a class="txt" href="#E177:16"><span class="lab"><font color="Green" title="E140">A153</font></span></a>, <a class="ref" href="jordan1j.html#T36" title="JORDAN1J:th.36">JORDAN1J:36</a></span>;<br/>
<a NAME="E216:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A246</font></span>: 
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font></span>)</span> or  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E166:16"><span class="lab"><font color="Green" title="E131">A143</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/>
<a NAME="E217:16"/><span class="lab"><font color="Green" title="E158">A247</font></span>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T61" title="JORDAN6:th.61">JORDAN6:61</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_41"><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="E1:16_41"/>
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font> or  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font> or  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E216:16"><span class="lab"><font color="Green" title="E157">A246</font></span></a>, <a class="ref" href="xboole_1.html#T70" title="XBOOLE_1:th.70">XBOOLE_1:70</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_41_1"><span class="kw">suppose </span></a><a NAME="E1:16_41_1"/>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">go</font>
 ; <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="E2:16_41_1"/><span class="kw">then </span>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:16"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E168:16"><span class="lab"><font color="Green" title="E133">A145</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_41_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E217:16"><span class="lab"><font color="Green" title="E158">A247</font></span></a>, <a class="ref" href="jordan10.html#T5" title="JORDAN10:th.5">JORDAN10:5</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</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="16_41_2"><span class="kw">suppose </span></a><a NAME="E1:16_41_2"/>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c26">pion1</font>
 ; <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"><span class="kw">hence </span><a NAME="E2:16_41_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E96:16"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E173:16"><span class="lab"><font color="Green" title="E138">A150</font></span></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="16_41_3"><span class="kw">suppose </span></a><a NAME="E1:16_41_3"/>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">do</font>
 ; <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="E2:16_41_3"/><span class="kw">then </span>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</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="c2">C</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:16"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E169:16"><span class="lab"><font color="Green" title="E134">A146</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_41_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E217:16"><span class="lab"><font color="Green" title="E158">A247</font></span></a>, <a class="ref" href="jordan10.html#T5" title="JORDAN10:th.5">JORDAN10:5</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</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>
