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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <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>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:68"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>;<br/>
<a NAME="E2:68"/><i><font color="Green">E38</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T33">REVROT_1:33</a></i>;<br/>
<a NAME="E3:68"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T14">REVROT_1:14</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<a NAME="E5:68"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:68"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E7:68"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E6:68"><i><font color="Green">E42</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E8:68"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E4:68"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:68"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E7:68"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan4.html#T49">JORDAN4:49</a></i>;<br/>
<a NAME="E11:68"/><i><font color="Green">E46</font></i>: 
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T25">JORDAN18:25</a></i>;<br/>
<a NAME="E12:68"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="ref" href="jordan18.html#T20">JORDAN18:20</a></i>;<br/>
<a NAME="E13:68"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T23">JORDAN18:23</a></i>;<br/>
<a NAME="E14:68"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E15:68"/><i><font color="Green">E50</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topreal3.html#T27">TOPREAL3:27</a></i>;<br/>
<a NAME="E16:68"/><i><font color="Green">E51</font></i>: 
(  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T22">JORDAN18:22</a></i>;<br/>
<a NAME="E17:68"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E16:68"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E18:68"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:68"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E20:68"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E17:68"><i><font color="Green">E52</font></i></a>, <a class="ref" href="goboard7.html#T38">GOBOARD7:38</a></i>;<br/>
<a NAME="E21:68"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_5.html#K2">:-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E19:68"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_5.html#T53">FINSEQ_5:53</a></i>;<br/>
<a NAME="E22:68"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E18:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_5.html#T42">FINSEQ_5:42</a></i>;<br/>
<a NAME="E23:68"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E17:68"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E25:68"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_5.html#K2">:-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E21:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>then </b> <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 = 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E19:68"><i><font color="Green">E54</font></i></a>, <a class="ref" href="revrot_1.html#T24">REVROT_1:24</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E18:68"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan23.html#E20:68"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_6.html#T52">FINSEQ_6:52</a></i>
;<br/>
<a NAME="E27:68"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="txt" href="jordan23.html#E16:68"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan3.html#T42">JORDAN3:42</a></i>;<br/>
<a NAME="E28:68"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<a NAME="E29:68"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:68"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E31:68"/><i><font color="Green">E59</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E32:68"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E33:68"/><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E34:68"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E9:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E35:68"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E4:68"><i><font color="Green">E40</font></i></a>, <a class="ref" href="revrot_1.html#T30">REVROT_1:30</a></i>;<br/>
<a NAME="E36:68"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E37:68"/><i><font color="Green">E65</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E5:68"><i><font color="Green">E41</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E38:68"/><i><font color="Green">E66</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E9:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan23.html#E6:68"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E39:68"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan23.html#E13:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E69</font></i>: 
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1"/>
( (  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V2">vertical</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V2">vertical</a> ) or (  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V2">vertical</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V1">horizontal</a> ) or (  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V1">horizontal</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V2">vertical</a> ) or (  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V1">horizontal</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V1">horizontal</a> ) )
 </span><b>by </b><i><a class="ref" href="sppol_1.html#T41">SPPOL_1:41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1"/><i><font color="Green">E70</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V2">vertical</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V2">vertical</a> )
 ;<br/><div class="add"><a NAME="E2:68_2_1_1"/><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E3:68_2_1_1"/><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T64">SPPOL_1:64</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1_1"/>
( <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="txt" href="jordan23.html#E39:68"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E2:68_2_1_1_1_1"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:68_2_1_1_1_1"/><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68_2_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E4:68_2_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <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="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E5:68_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E6:68_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:68_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="jordan23.html#E30:68"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E8:68_2_1_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><a NAME="E9:68_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_1_1_1_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/>

<a NAME="E2:68_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
;<br/>
<a NAME="E3:68_2_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan23.html#E4:68_2_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E4:68_2_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E33:68"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan23.html#E35:68"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:68_2_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:68_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E6:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan23.html#E9:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan23.html#E8:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan23.html#E34:68"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan23.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:68_2_1_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E11:68_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:68_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E13:68_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E8:68_2_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:68_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/><b>hence </b><a NAME="E15:68_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan23.html#E10:68_2_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E2:68_2_1_1_1_2"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:68_2_1_1_1_2"/><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68_2_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E4:68_2_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E5:68_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E6:68_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:68_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E8:68_2_1_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><a NAME="E9:68_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2_1_1_1_2_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/>

<a NAME="E2:68_2_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
;<br/>
<a NAME="E3:68_2_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_1_1_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan23.html#E4:68_2_1_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E4:68_2_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E33:68"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan23.html#E35:68"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:68_2_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 or <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:68_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E6:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan23.html#E9:68"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan23.html#E8:68"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan23.html#E34:68"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan23.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:68_2_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E11:68_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:68_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E2:68_2_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/><a NAME="E13:68_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E8:68_2_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E14:68_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/><b>hence </b><a NAME="E15:68_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan23.html#E3:68_2_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan23.html#E10:68_2_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_2"/><i><font color="Green">E70</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V2">vertical</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V1">horizontal</a> )
 ;<br/><div class="add"><a NAME="E2:68_2_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T64">SPPOL_1:64</a></i>;<br/><a NAME="E3:68_2_1_2"/>
<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T63">SPPOL_1:63</a></i>;<br/><a NAME="E4:68_2_1_2"/><b>then </b>
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/><a NAME="E5:68_2_1_2"/><b>then </b>
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E6:68_2_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E7:68_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><a NAME="E8:68_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:68_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/><b>hence </b><a NAME="E10:68_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E13:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_3"><b>suppose </b></a><a NAME="E1:68_2_1_3"/><i><font color="Green">E70</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V1">horizontal</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V2">vertical</a> )
 ;<br/><div class="add"><a NAME="E2:68_2_1_3"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E12:68"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T64">SPPOL_1:64</a></i>;<br/><a NAME="E3:68_2_1_3"/>
<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sppol_1.html#T63">SPPOL_1:63</a></i>;<br/><a NAME="E4:68_2_1_3"/><b>then </b>
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="jordan23.html#E2:68_2_1_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/><a NAME="E5:68_2_1_3"/><b>then </b>
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E32:68"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E6:68_2_1_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E7:68_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><a NAME="E8:68_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:68_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/><b>hence </b><a NAME="E10:68_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E13:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_4"><b>suppose </b></a><a NAME="E1:68_2_1_4"/><i><font color="Green">E70</font></i>: 
(  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1 is <a href="sppol_1.html#V1">horizontal</a> &amp;  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> is <a href="sppol_1.html#V1">horizontal</a> )
 ;<br/><div class="add"><a NAME="E2:68_2_1_4"/>
( <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sppol_1.html#T63">SPPOL_1:63</a></i>;<br/><b>hence </b><a NAME="E3:68_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E37:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan23.html#E38:68"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan23.html#E36:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan23.html#E13:68"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan23.html#E1:68_2_1_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="sprect_3.html#T21">SPRECT_3:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E41:68"/><i><font color="Green">E69</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E6:68"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan23.html#E9:68"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard2.html#T8">GOBOARD2:8</a></i>;<br/>
<a NAME="E42:68"/><i><font color="Green">E70</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>14</sub></font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E43:68"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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="finseq_6.html#K1">Rotate</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E44:68"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E45:68"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan23.html#E7:68"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E46:68"/><b>then </b>
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>14</sub></font>,1
 
<b>by </b><i><a class="txt" href="jordan23.html#E27:68"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sppol_2.html#T3">SPPOL_2:3</a></i>;<br/>
<a NAME="E47:68"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E16:68"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan23.html#E2:68"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan23.html#E40:68"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan23.html#E41:68"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan23.html#E42:68"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="jordan23.html#E11:68"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T72">JORDAN3:72</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>;<br/>
<a NAME="E49:68"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E49:68"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/>
<a NAME="E51:68"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E52:68"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E53:68"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E54:68"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E55:68"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E56:68"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b><i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T61">GRAPH_2:61</a></i>
<br/>.= 
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:68_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 ;<br/><a NAME="E2:68_5"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_5_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan23.html#E2:68_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_5_1_1"><b>suppose </b></a><a NAME="E1:68_5_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:68_5_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E56:68"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan23.html#E52:68"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:68_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:68_5_1_1"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:68_5_1_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>thus </b><a NAME="E7:68_5_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E5:68_5_1_1"><i><font color="Green">E81</font></i></a></i>;<br/><b>thus </b><a NAME="E8:68_5_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E6:68_5_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan23.html#E1:68_5_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan23.html#E57:68"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_5_1_2"><b>suppose </b></a><a NAME="E1:68_5_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:68_5_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E51:68"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan23.html#E53:68"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:68_5_1_2"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:68_5_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>thus </b><a NAME="E6:68_5_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E4:68_5_1_2"><i><font color="Green">E81</font></i></a></i>;<br/><b>thus </b><a NAME="E7:68_5_1_2"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E49:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E5:68_5_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan23.html#E1:68_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="graph_2.html#T62">GRAPH_2:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E59:68"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 
<b>by </b><i><a class="txt" href="jordan23.html#E11:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan23.html#E31:68"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan1b.html#T8">JORDAN1B:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM8">FinSequence</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="E61:68"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E62:68"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E63:68"/><i><font color="Green">E81</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E64:68"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E65:68"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E66:68"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E58:68"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard3.html#T2">GOBOARD3:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="jordan23.html#E62:68"><i><font color="Green">E80</font></i></a></i>;<br/>
<div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:68_6"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:68_6"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan23.html#T23" target="_self">Th23</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:68_6"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:68_6"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/><b>thus </b><a NAME="E6:68_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E4:68_6"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E7:68_6"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:68_6"/>
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p3"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E5:68_6"><i><font color="Green">E88</font></i></a>, <a class="ref" href="graph_2.html#T61">GRAPH_2:61</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM8">FinSequence</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="E70:68"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E71:68"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E72:68"/><i><font color="Green">E88</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E73:68"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E74:68"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E75:68"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E68:68"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard3.html#T2">GOBOARD3:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="jordan23.html#E71:68"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<i><font color="Green">E92</font></i>: <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> = 
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E74:68"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan23.html#E64:68"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E78:68"/><i><font color="Green">E93</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_8"/>
<font color="Maroon">c<sub>21</sub></font> is <a href="funct_1.html#V3">constant</a>
 ;<br/><a NAME="E2:68_8"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E78:68"><i><font color="Green">E93</font></i></a>, <a class="ref" href="revrot_1.html#D1">REVROT_1:def 1</a></i>;<br/><a NAME="E3:68_8"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E73:68"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jordan23.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E4:68_8"/><b>then </b>
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E74:68"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E5:68_8"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T25">JORDAN18:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E80:68"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>19</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E61:68"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan23.html#E77:68"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan23.html#E70:68"><i><font color="Green">E86</font></i></a>, <a class="ref" href="topreal8.html#T12">TOPREAL8:12</a></i>;<br/>
<a NAME="E81:68"/><i><font color="Green">E96</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E72:68"><i><font color="Green">E88</font></i></a>, <a class="ref" href="jordan5b.html#T27">JORDAN5B:27</a></i>;<br/>
<a NAME="E82:68"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E83:68"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E81:68"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E84:68"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E15:68"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E85:68"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E63:68"><i><font color="Green">E81</font></i></a>, <a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E86:68"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T21">SPPOL_2:21</a></i>;<br/>
<a NAME="E87:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E84:68"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E88:68"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/>
<a NAME="E89:68"/><i><font color="Green">E99</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E90:68"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E91:68"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E90:68"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E92:68"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E89:68"><i><font color="Green">E99</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E93:68"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E94:68"/>
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E95:68"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E74:68"><i><font color="Green">E90</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E96:68"/><i><font color="Green">E104</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E15:68"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan23.html#E81:68"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_9"/>
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><a NAME="E2:68_9"/><b>then </b>
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E91:68"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan23.html#E93:68"><i><font color="Green">E102</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:68_9"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E95:68"><i><font color="Green">E103</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/><a NAME="E4:68_9"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E5:68_9"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E91:68"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan23.html#E93:68"><i><font color="Green">E102</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E6:68_9"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/><a NAME="E7:68_9"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E14:68"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan23.html#E96:68"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:68_9"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/><b>hence </b><a NAME="E9:68_9"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E13:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E98:68"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_10"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:68_10_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span>
 ;<br/>

<a NAME="E2:68_10_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E97:68"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan23.html#E1:68_10_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:68_10_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E88:68"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan23.html#E1:68_10_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:68_10_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E74:68"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:68_10_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:68_10"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="realset1.html#K5">}</a></span>
 ;<br/>

<a NAME="E3:68_10"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:68_10"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68_10"><i><font color="Green">E107</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:68_10"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E91:68"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan23.html#E93:68"><i><font color="Green">E102</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E6:68_10"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:68_10"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan23.html#E77:68"><i><font color="Green">E92</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E7:68_10"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1
 
<b>by </b><i><a class="txt" href="jordan23.html#E49:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E66:68"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:68_10"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>19</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E5:68_10"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E107</font></i>: <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> = 
 <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E49:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E65:68"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E73:68"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jordan23.html#T19" target="_self">Th19</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_12"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_12"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:68_12"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan23.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:68_12"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E11:68"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan23.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E101:68"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E56:68"><i><font color="Green">E76</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E102:68"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E75:68"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E103:68"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E104:68"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:68"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan18.html#T27">JORDAN18:27</a></i>;<br/>
<a NAME="E105:68"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E15:68"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan23.html#E81:68"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E106:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E104:68"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E107:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E63:68"><i><font color="Green">E81</font></i></a>, <a class="ref" href="sppol_2.html#T21">SPPOL_2:21</a></i>;<br/>
<a NAME="E108:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E47:68"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan23.html#E73:68"><i><font color="Green">E89</font></i></a>, <a class="ref" href="jordan23.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E109:68"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E64:68"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E110:68"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:68_13"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:68_13"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:68_13"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan23.html#E99:68"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan23.html#E103:68"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan23.html#E49:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan23.html#E66:68"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan23.html#E77:68"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/>
<b>hence </b><a NAME="E4:68_13"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E111:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E109:68"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>19</sub></font> as  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> <b>by </b><i><a class="txt" href="jordan23.html#E79:68"><i><font color="Green">E94</font></i></a>, <a class="txt" href="jordan23.html#E80:68"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan23.html#E98:68"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan23.html#E99:68"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan23.html#E77:68"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jordan23.html#T25" target="_self">Th25</a>, <a class="ref" href="topreal8.html#T11">TOPREAL8:11</a>, <a class="ref" href="topreal8.html#T33">TOPREAL8:33</a>, <a class="ref" href="topreal8.html#T34">TOPREAL8:34</a>, <a class="ref" href="jordan8.html#T7">JORDAN8:7</a></i>;<br/>
<a NAME="E113:68"/><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>thus </b><a NAME="E114:68"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K5">B_Cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_6.html#K1">Rotate</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>thus </b><a NAME="E115:68"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan23.html#E61:68"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan23.html#E63:68"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan23.html#E64:68"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan23.html#E65:68"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan23.html#E66:68"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan23.html#E113:68"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/>

<b>thus </b><a NAME="E116:68"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="graph_2.html#K4">^'</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="jordan18.html#K1">North-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan18.html#K2">South-Bound</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="jordan23.html#E70:68"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan23.html#E72:68"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan23.html#E73:68"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan23.html#E74:68"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan23.html#E75:68"><i><font color="Green">E91</font></i></a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>thus </b><a NAME="E117:68"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>


</div>
