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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="connsp_1.html#V2">connected</a> <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>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pscomp_1.html#K29">N-min</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>4</sub></font> =  <a href="pscomp_1.html#K30">N-max</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>5</sub></font> =  <a href="pscomp_1.html#K27">W-min</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>6</sub></font> =  <a href="pscomp_1.html#K28">W-max</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>7</sub></font> =  <a href="pscomp_1.html#K31">E-max</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>8</sub></font> =  <a href="pscomp_1.html#K32">E-min</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>9</sub></font> =  <a href="pscomp_1.html#K33">S-max</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>10</sub></font> =  <a href="pscomp_1.html#K34">S-min</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>11</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"><a href="pscomp_1.html#K27">W-min</a> <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>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"><a href="pscomp_1.html#K31">E-max</a> <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/>
<a NAME="E1:5"/><i><font color="Green">E5</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> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</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="ref" href="jordan9.html#T34">JORDAN9:34</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green">E6</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#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="sprect_2.html#T43">SPRECT_2:43</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green">E7</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#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="sprect_2.html#T47">SPRECT_2:47</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E8</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#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="sprect_2.html#T50">SPRECT_2:50</a></i>;<br/>
<a NAME="E5:5"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T72">SPRECT_2:72</a></i>;<br/>
<a NAME="E6:5"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <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="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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T74">SPRECT_2:74</a></i>;<br/>
<a NAME="E7:5"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E5:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:5"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T75">SPRECT_2:75</a></i>;<br/>
<a NAME="E9:5"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <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="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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T76">SPRECT_2:76</a></i>;<br/>
<a NAME="E10:5"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T77">SPRECT_2:77</a></i>;<br/>
<a NAME="E11:5"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <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="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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="sprect_2.html#T78">SPRECT_2:78</a></i>;<br/>
<a NAME="E12:5"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E8:5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan1j.html#E9:5"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E10:5"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:5"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E11:5"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:5"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_5.html#T49">FINSEQ_5:49</a></i>;<br/>
<a NAME="E16:5"/><i><font color="Green">E18</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_5.html#K1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T50">FINSEQ_5:50</a></i>;<br/>
<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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_4.html#K4">/.</a> 1 = 
<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> 1
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T47">FINSEQ_5:47</a></i>
<br/>.= 
 <a href="pscomp_1.html#K29">N-min</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="ref" href="jordan9.html#T34">JORDAN9:34</a></i>
;<br/>
<a NAME="E18:5"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E16:5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<a NAME="E19:5"/>
 <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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T45">FINSEQ_5:45</a></i>;<br/>
<a NAME="E20:5"/><b>then </b>
<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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="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"><span class="p3">(<span class="default"><a href="jordan9.html#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#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</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="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T48">FINSEQ_5:48</a></i>;<br/>
<a NAME="E21:5"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E16:5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<a NAME="E22:5"/>
(  <a href="pscomp_1.html#K28">W-max</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#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; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.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="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/>
<a NAME="E23:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <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="pscomp_1.html#K29">N-min</a> <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="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E24:5"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</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="ref" href="sprect_2.html#T61">SPRECT_2:61</a></i>;<br/>
<a NAME="E25:5"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<a NAME="E26:5"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:5_2"/>
<font color="Maroon">c<sub>13</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="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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>
 ;<br/>

<b>hence </b><a NAME="E2:5_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 <b>by </b><i><a class="txt" href="jordan1j.html#E18:5"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1j.html#E21:5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:5"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E28:5"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E29:5"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E30:5"/><b>then </b>
2 <a href="tarski.html#R1">c=</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E25:5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1j.html#E27:5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:5"/><b>then </b>
 <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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E32:5"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>  <a href="topreal1.html#K5">L~</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/>
<a NAME="E33:5"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="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#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E7:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1j.html#E14:5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:5"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T49">FINSEQ_5:49</a></i>;<br/>
<a NAME="E35:5"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_4.html#K5">..</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_3"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_4.html#K5">..</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="hidden.html#R1">=</a> 1
 ;<br/>

<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="finseq_4.html#K5">..</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="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>
<b>by </b><i><a class="txt" href="jordan1j.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan1j.html#E33:5"><i><font color="Green">E25</font></i></a>, <a class="ref" href="sprect_5.html#T3">SPRECT_5:3</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="jordan1j.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_6.html#T47">FINSEQ_6:47</a></i>
;<br/>
<b>hence </b><a NAME="E3:5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1j.html#E5:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan1j.html#E6:5"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan1j.html#E15:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan1j.html#E34:5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan1j.html#E1:5_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_5.html#T10">FINSEQ_5:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:5"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E15:5"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_6.html#T83">FINSEQ_6:83</a></i>;<br/>
<a NAME="E37:5"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <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> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T55">SPRECT_2:55</a></i>;<br/>
<a NAME="E38:5"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <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="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <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> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T97">PSCOMP_1:97</a></i>;<br/>
<a NAME="E39:5"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <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="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.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="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E40:5"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</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="jordan1j.html#E37:5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E41:5"/>
not  <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E31</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 ;<br/>

<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"><a href="pscomp_1.html#K27">W-min</a> <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="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"><span class="p3">(<span class="default"><a href="jordan9.html#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="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</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="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="jordan9.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="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_5.html#T57">FINSEQ_5:57</a></i>
<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> 1
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>
<br/>.= 
 <a href="pscomp_1.html#K29">N-min</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="ref" href="jordan9.html#T34">JORDAN9:34</a></i>
;<br/>
<a NAME="E3:5_4"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<a NAME="E4:5_4"/>
<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"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</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="ref" href="finseq_5.html#T56">FINSEQ_5:56</a></i>;<br/>
<a NAME="E5:5_4"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<a NAME="E6:5_4"/>
(  <a href="pscomp_1.html#K28">W-max</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#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; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.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="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/>
<a NAME="E7:5_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-max</a> <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="pscomp_1.html#K29">N-min</a> <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="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E8:5_4"/><b>then </b>
 <a href="pscomp_1.html#K29">N-min</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#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</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="ref" href="sprect_2.html#T61">SPRECT_2:61</a></i>;<br/>
<a NAME="E9:5_4"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<a NAME="E10:5_4"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:5_4_2"/>
<font color="Maroon">c<sub>13</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="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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>
 ;<br/>

<b>hence </b><a NAME="E2:5_4_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 <b>by </b><i><a class="txt" href="jordan1j.html#E3:5_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1j.html#E5:5_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:5_4"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E12:5_4"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E13:5_4"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E14:5_4"/><b>then </b>
2 <a href="tarski.html#R1">c=</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E9:5_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1j.html#E11:5_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:5_4"/><b>then </b>
 <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"><a href="pscomp_1.html#K27">W-min</a> <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="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E16:5_4"/><b>then </b>
 <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K27">W-min</a> <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>  <a href="topreal1.html#K5">L~</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"><a href="pscomp_1.html#K27">W-min</a> <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>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/>
<a NAME="E17:5_4"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</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"><span class="p3">(<span class="default"><a href="jordan9.html#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="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E15:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan1j.html#E32:5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan1j.html#E1:5_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:5_4"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K27">W-min</a> <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="ref" href="jordan1g.html#T25">JORDAN1G:25</a></i>;<br/>
<a NAME="E19:5_4"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</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>  <a href="pscomp_1.html#K27">W-min</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="jordan1j.html#E40:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:5_4"/>
contradiction
 <b>by </b><i><a class="ref" href="topreal5.html#T25">TOPREAL5:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:5"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E36:5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E43:5"/><i><font color="Green">E32</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan1j.html#E14:5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_6.html#T67">FINSEQ_6:67</a></i>;<br/>
<a NAME="E44:5"/><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="jordan1j.html#E40:5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<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"><a href="pscomp_1.html#K31">E-max</a> <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="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"><span class="p3">(<span class="default"><a href="jordan9.html#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="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</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="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="jordan9.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="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_5.html#T57">FINSEQ_5:57</a></i>
<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> 1
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>
<br/>.= 
 <a href="pscomp_1.html#K29">N-min</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="ref" href="jordan9.html#T34">JORDAN9:34</a></i>
;<br/>
<a NAME="E46:5"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<a NAME="E47:5"/>
<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"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</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="ref" href="finseq_5.html#T56">FINSEQ_5:56</a></i>;<br/>
<a NAME="E48:5"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<a NAME="E49:5"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:5_6"/>
<font color="Maroon">c<sub>13</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="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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>
 ;<br/>

<b>hence </b><a NAME="E2:5_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 <b>by </b><i><a class="txt" href="jordan1j.html#E46:5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1j.html#E48:5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:5"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E51:5"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E52:5"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E53:5"/><b>then </b>
2 <a href="tarski.html#R1">c=</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E44:5"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1j.html#E50:5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E54:5"/><b>then </b>
 <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"><a href="pscomp_1.html#K31">E-max</a> <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="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E55:5"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</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"><a href="pscomp_1.html#K31">E-max</a> <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>  <a href="topreal1.html#K5">L~</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"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/>
<a NAME="E56:5"/>
not  <a href="pscomp_1.html#K27">W-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_7"/><i><font color="Green">E38</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 ;<br/>

<a NAME="E2:5_7"/><b>then </b><i><font color="Green">E39</font></i>: 
not <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_5.html#K1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_4.html#K4">/.</a> 1 = 
<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> 1
<b>by </b><i><a class="txt" href="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_5.html#T47">FINSEQ_5:47</a></i>
<br/>.= 
 <a href="pscomp_1.html#K29">N-min</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="ref" href="jordan9.html#T34">JORDAN9:34</a></i>
;<br/>
<a NAME="E4:5_7"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="pscomp_1.html#K29">N-min</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:5_7"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_6.html#T46">FINSEQ_6:46</a></i>;<br/>
<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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="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"><span class="p3">(<span class="default"><a href="jordan9.html#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#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></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_5.html#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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> <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>
<b>by </b><i><a class="txt" href="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_5.html#T45">FINSEQ_5:45</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</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="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_5.html#T48">FINSEQ_5:48</a></i>
;<br/>
<a NAME="E6:5_7"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E2:5_7"><i><font color="Green">E39</font></i></a>, <a class="ref" href="revrot_1.html#T3">REVROT_1:3</a></i>;<br/>
<a NAME="E7:5_7"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:5_7_3"/>
<font color="Maroon">c<sub>13</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="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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>
 ;<br/>

<b>hence </b><a NAME="E2:5_7_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 <b>by </b><i><a class="txt" href="jordan1j.html#E4:5_7"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1j.html#E6:5_7"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:5_7"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E9:5_7"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E10:5_7"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span> <a href="tarski.html#R1">c=</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E11:5_7"/><b>then </b>
2 <a href="tarski.html#R1">c=</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E44:5"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1j.html#E8:5_7"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:5_7"/><b>then </b>
 <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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E13:5_7"/><b>then </b>
 <a href="relat_1.html#K2">rng</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>  <a href="topreal1.html#K5">L~</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#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18">SPPOL_2:18</a></i>;<br/>
<a NAME="E14:5_7"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</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"><span class="p3">(<span class="default"><a href="jordan9.html#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="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E43:5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1j.html#E55:5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1j.html#E1:5_7"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:5_7"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <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="pscomp_1.html#K31">E-max</a> <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="ref" href="jordan1j.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E16:5_7"/><b>then </b>
 <a href="pscomp_1.html#K27">W-min</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>  <a href="pscomp_1.html#K31">E-max</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="jordan1j.html#E24:5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E17:5_7"/>
contradiction
 <b>by </b><i><a class="ref" href="topreal5.html#T25">TOPREAL5:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E57:5"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="pscomp_1.html#K27">W-min</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#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<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"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_5.html#K1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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"><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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="finseq_1.html#K8">^</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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finseq_6.html#D2">FINSEQ_6:def 2</a></i>
<br/>.= 
<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"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>
<b>by </b><i><a class="txt" href="jordan1j.html#E42:5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_6.html#T72">FINSEQ_6:72</a></i>
<br/>.= 
<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"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan1j.html#E15:5"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan1j.html#E35:5"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_6.html#T65">FINSEQ_6:65</a></i>
<br/>.= 
<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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="finseq_5.html#K2">:-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan1j.html#E57:5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_6.html#T76">FINSEQ_6:76</a></i>
<br/>.= 
<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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="finseq_5.html#K2">:-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="finseq_1.html#K8">^</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_5.html#K1">-:</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan1j.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan1j.html#E15:5"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_6.html#T80">FINSEQ_6:80</a></i>
<br/>.= 
<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_5.html#K2">:-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="finseq_1.html#K8">^</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_5.html#K1">-:</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <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>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="finseq_5.html#K2">:-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="jordan1j.html#E43:5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_6.html#T69">FINSEQ_6:69</a></i>
<br/>.= 
<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"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_5.html#K2">:-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="jordan1j.html#E4:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_6.html#D2">FINSEQ_6:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E59:5"/>
 <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <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"><a href="pscomp_1.html#K31">E-max</a> <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="finseq_5.html#K2">:-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <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="jordan1e.html#D1">JORDAN1E:def 1</a></i>;<br/>


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