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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V21</a>()  <a href="goboard1.html#V5" title="GOBOARD1:attr.5">X_increasing-in-column</a>  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span><br/>

<a NAME="E1:25"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E2:25"/><span class="kw">then </span>
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> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E3:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,1)</span>)</span> as   <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D7" title="GOBOARD1:def.7">GOBOARD1:def 7</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A2</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">L</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E20">A3</font></span>:  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">L</font> = 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T10" target="_self" title="JORDAN1H:th.10">Th16</a></span>
<br/>.= 
<a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T127" title="RELAT_1:th.127">RELAT_1:127</a></span>
;<br/>
<a NAME="E7:25"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<font color="Maroon" title="c1">G</font>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="jordan1h.html#T12" target="_self" title="JORDAN1H:th.12">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:25"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</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/>


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