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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:67"/><i><font color="Green">E37</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:67"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:67"/><i><font color="Green">E39</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:67"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E2:67"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard6.html#T29">GOBOARD6:29</a></i>;<br/>
<b>assume </b><a NAME="E4:67"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:67"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:67"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:67"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E10:67"/><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E11:67"/><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:67"><i><font color="Green">E39</font></i></a>, <a class="txt" href="sprect_3.html#E6:67"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E12:67"/><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd11.html#T34">GOBRD11:34</a></i>;<br/>
<a NAME="E13:67"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:67"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sprect_3.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E14:67"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd11.html#T34">GOBRD11:34</a></i>;<br/>
<a NAME="E15:67"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:67"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sprect_3.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E16:67"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:67"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:67"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E12:67"><i><font color="Green">E45</font></i></a>, <a class="ref" href="sprect_3.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E18:67"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:67"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:67"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E14:67"><i><font color="Green">E47</font></i></a>, <a class="ref" href="sprect_3.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E20:67"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : ( ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="sprect_3.html#T52" target="_self">Th52</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E22:67"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E23:67"/><i><font color="Green">E52</font></i>: 
( ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="sprect_3.html#E7:67"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E24:67"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E9:67"><i><font color="Green">E42</font></i></a>, <a class="txt" href="sprect_3.html#E22:67"><i><font color="Green">E51</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E25:67"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E9:67"><i><font color="Green">E42</font></i></a>, <a class="txt" href="sprect_3.html#E22:67"><i><font color="Green">E51</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E26:67"/><i><font color="Green">E55</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T66">SPRECT_1:66</a>, <a class="ref" href="sprect_1.html#T67">SPRECT_1:67</a>, <a class="ref" href="sprect_1.html#T68">SPRECT_1:68</a>, <a class="ref" href="sprect_1.html#T69">SPRECT_1:69</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E23:67"><i><font color="Green">E52</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1
 <b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><a NAME="E3:67_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>hence </b><a NAME="E4:67_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E10:67"><i><font color="Green">E43</font></i></a>, <a class="txt" href="sprect_3.html#E13:67"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sprect_3.html#E24:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sprect_3.html#E26:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E1:67_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sprect_3.html#E2:67_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_2"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><a NAME="E3:67_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>hence </b><a NAME="E4:67_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E11:67"><i><font color="Green">E44</font></i></a>, <a class="txt" href="sprect_3.html#E19:67"><i><font color="Green">E50</font></i></a>, <a class="txt" href="sprect_3.html#E25:67"><i><font color="Green">E54</font></i></a>, <a class="txt" href="sprect_3.html#E26:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E1:67_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sprect_3.html#E2:67_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3"><b>suppose </b></a><a NAME="E1:67_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_3"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><a NAME="E3:67_1_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>hence </b><a NAME="E4:67_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E11:67"><i><font color="Green">E44</font></i></a>, <a class="txt" href="sprect_3.html#E15:67"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E25:67"><i><font color="Green">E54</font></i></a>, <a class="txt" href="sprect_3.html#E26:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E1:67_1_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sprect_3.html#E2:67_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_4"><b>suppose </b></a><a NAME="E1:67_1_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_1_4"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
 <b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><a NAME="E3:67_1_4"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>hence </b><a NAME="E4:67_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E10:67"><i><font color="Green">E43</font></i></a>, <a class="txt" href="sprect_3.html#E17:67"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sprect_3.html#E24:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="sprect_3.html#E26:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E1:67_1_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sprect_3.html#E2:67_1_4"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
