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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="relat_1.html#V3">V6</a> <a href="goboard1.html#V5">Y_increasing-in-line</a> <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:30"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E2:30"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E21</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="goboard1.html#K3">Y_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> as  <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="goboard1.html#D8">GOBOARD1:def 8</a></i>;<br/>
<i><font color="Green">E22</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1h.html#T17" target="_self">Th17</a></i>
<br/>.= 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T160">RELAT_1:160</a></i>
;<br/>
<i><font color="Green">E23</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>
<b>by </b><i><a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E7:30"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E3:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan1h.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E8:30"/><b>then </b>
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E9:30"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E5:30"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1h.html#E6:30"><i><font color="Green">E23</font></i></a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>


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