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







<b>assume </b><b>that </b><br/><a NAME="E1:50"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E3:50"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E4:50"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T7">HEINE:7</a></i>;<br/>
<a NAME="E7:50"/><b>then </b><i><font color="Green">E39</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E5:50"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E6:50"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E9:50"/>
1 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan1h.html#E2:50"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:50"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E8:50"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E11:50"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E7:50"><i><font color="Green">E39</font></i></a>, <a class="ref" href="real_2.html#T203">REAL_2:203</a></i>;<br/>
<a NAME="E12:50"/><b>then </b>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E10:50"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:50"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:50"/><b>then </b>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan1h.html#E4:50"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:50"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 2</span>)</span></span><a href="int_1.html#K1">/]</a></span> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>


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