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<b>let </b><font color="Maroon">f</font> be  <a href="sprect_1.html#V1">rectangular</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 ;<br/>

<b>set </b><font color="Maroon">lp</font> =  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">ilp</font> =  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:60"/><i><font color="Green">E37</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:60"/><i><font color="Green">E38</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:60"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E2:60"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E5:60"/><i><font color="Green">E63</font></i>: 
(  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T49">SPPOL_1:49</a></i>;<br/>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E7:60"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T31">TOPREAL1:31</a></i>;<br/>
<a NAME="E8:60"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:60"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jordan5c.html#D2">JORDAN5C:def 2</a></i>;<br/>
<a NAME="E9:60"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:60"/><i><font color="Green">E67</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:60"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E12:60"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:60"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:60"/><i><font color="Green">E70</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan3.html#T42">JORDAN3:42</a></i>;<br/>
<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E15:60"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E16:60"/><i><font color="Green">E87</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> or  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D1">SPRECT_2:def 1</a><br/>

<a NAME="E17:60"/>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  not  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> or  not  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E18:60"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E19:60"/><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:60"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E21:60"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E22:60"/><i><font color="Green">E94</font></i>: 
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E23:60"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E24:60"/><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then </b><i><font color="Green">E97</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="jordan4.html#T20">JORDAN4:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_2.html#T9">NAT_2:9</a></i>
;<br/>
<a NAME="E26:60"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E27:60"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="sprect_3.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:60"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:60"/><i><font color="Green">E101</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T96">SPRECT_1:96</a></i>;<br/>
<a NAME="E30:60"/><i><font color="Green">E102</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2_1"/>
(  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> or  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_1"><b>suppose </b></a><a NAME="E1:60_2_1_1"/>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:60_2_1_1"/><b>then </b>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><a NAME="E3:60_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>;<br/><b>hence </b><a NAME="E4:60_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:60"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sprect_3.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1_2"><b>suppose </b></a><a NAME="E1:60_2_1_2"/><i><font color="Green">E104</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:60_2_1_2"/><b>then </b>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , , <a class="ref" href="sprect_3.html#T14" target="_self">Th14</a>, , <a class="ref" href="sprect_2.html#T7">SPRECT_2:7</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_3.html#T14" target="_self">Th14</a>, <a class="txt" href="sprect_3.html#E5:60"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:60_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:60"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sprect_3.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E32:60"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E33:60"/><b>then </b><i><font color="Green">E114</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E34:60"/><b>then </b><i><font color="Green">E115</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E35:60"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>;<br/>
<a NAME="E36:60"/><i><font color="Green">E117</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_3"/><i><font color="Green">E118</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:60_3"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><a NAME="E3:60_3"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E4:60_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E5:60_3"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</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="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E6:60_3"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="sprect_3.html#E9:60"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a></i>;<br/><a NAME="E7:60_3"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E8:60_3"/>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:60"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sprect_3.html#E14:60"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E134</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="sprect_3.html#T14" target="_self">Th14</a>, <a class="txt" href="sprect_3.html#E15:60"><i><font color="Green">E71</font></i></a>, <a class="ref" href="sprect_2.html#T7">SPRECT_2:7</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i/>
;<br/><a NAME="E10:60_3"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E11:60_3"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:60"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="sprect_3.html#T16" target="_self">Th16</a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">m</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i>, <a class="txt" href="sprect_3.html#E6:60"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="jordan3.html#T39">JORDAN3:39</a></i>;<br/><a NAME="E13:60_3"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:60"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:60_3"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">m</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , , <a class="ref" href="sprect_2.html#T12">SPRECT_2:12</a></i>;<br/><a NAME="E15:60_3"/>
<font color="Maroon">m</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:60"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="sprect_2.html#T13">SPRECT_2:13</a></i>;<br/><a NAME="E16:60_3"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="sprect_3.html#T19" target="_self">Th19</a>, </i>;<br/><b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:60_3"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E19:60_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:60_3"/><i><font color="Green">E141</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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> <font color="Maroon">m</font> )
 <b>and </b><br/><a NAME="E23:60_3"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">m</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/><a NAME="E24:60_3"/><i><font color="Green">E143</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:60"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="jordan4.html#T20">JORDAN4:20</a></i>;<br/><a NAME="E25:60_3"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E26:60_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E27:60_3"/><b>then </b><i><font color="Green">E145</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">m</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:60"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="sprect_3.html#T27" target="_self">Th27</a>, <a class="ref" href="sprect_3.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan4.html#T31">JORDAN4:31</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E28:60_3"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>;<br/><a NAME="E29:60_3"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E30:60_3"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="sprect_3.html#E8:60"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sprect_3.html#E9:60"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E31:60_3"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E32:60_3"/><i><font color="Green">E148</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="sprect_3.html#T14" target="_self">Th14</a>, <a class="txt" href="sprect_3.html#E10:60"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E33:60_3"/><i><font color="Green">E149</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="sprect_3.html#T13" target="_self">Th13</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E34:60_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T27" target="_self">Th27</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E35:60_3"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/><a NAME="E36:60_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E37:60_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E38:60_3"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E19:60"><i><font color="Green">E89</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_3_2"/>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:60_3_2"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T26" target="_self">Th26</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:60_3_2"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_3_2_1"/>
( <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 or <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_2_1_1"><b>suppose </b></a><a NAME="E1:60_3_2_1_1"/><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:60_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E20:60"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E3:60_3_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:60_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E16:60"><i><font color="Green">E87</font></i></a>, , <a class="txt" href="sprect_3.html#E21:60"><i><font color="Green">E93</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_3_2_1_2"><b>suppose </b></a><a NAME="E1:60_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_3_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E40:60_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="sprect_3.html#E4:60"><i><font color="Green">E38</font></i></a>, , , , , <a class="ref" href="sprect_3.html#T20" target="_self">Th20</a>, <a class="ref" href="sprect_3.html#T26" target="_self">Th26</a>, , <a class="txt" href="sprect_3.html#E18:60"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_3.html#E20:60"><i><font color="Green">E90</font></i></a>, <a class="ref" href="jordan5c.html#T28">JORDAN5C:28</a></i>;<br/><a NAME="E41:60_3"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E42:60_3"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E38:60"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/>
<a NAME="E39:60"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E40:60"/><i><font color="Green">E157</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:60"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:60"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E42:60"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>;<br/>
<a NAME="E43:60"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan3.html#T26">JORDAN3:26</a></i>;<br/>
<a NAME="E44:60"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:60"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E14:60"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan4.html#T18">JORDAN4:18</a></i>;<br/>
<i><font color="Green">E172</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , , <a class="ref" href="sprect_3.html#T14" target="_self">Th14</a>, <a class="txt" href="sprect_3.html#E10:60"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sprect_2.html#T7">SPRECT_2:7</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<a NAME="E46:60"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:60"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="sprect_3.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i>, , <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan3.html#T39">JORDAN3:39</a></i>;<br/>
<a NAME="E48:60"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E49:60"/><b>then </b>
<font color="Maroon">m</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T7" target="_self">Th7</a>, , <a class="ref" href="sprect_3.html#T17" target="_self">Th17</a>, <a class="ref" href="sprect_2.html#T12">SPRECT_2:12</a></i>;<br/>
<a NAME="E50:60"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E15:60"><i><font color="Green">E71</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:60"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E53:60"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="sprect_3.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E56:60"/><i><font color="Green">E176</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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> <font color="Maroon">m</font> )
 <b>and </b><br/><a NAME="E57:60"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">m</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="sprect_3.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E58:60"/><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E59:60"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="txt" href="sprect_3.html#E11:60"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E60:60"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E61:60"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">m</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="sprect_3.html#T20" target="_self">Th20</a>, <a class="ref" href="jordan4.html#T31">JORDAN4:31</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E62:60"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>;<br/>
<a NAME="E63:60"/><b>then </b><i><font color="Green">E201</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E64:60"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E65:60"/><b>then </b><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="sprect_3.html#T27" target="_self">Th27</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E203</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_5"/>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:60_5"/><b>then </b>
 <a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p0">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sprect_3.html#T19" target="_self">Th19</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:60_5"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E4:60_5"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_3.html#T28" target="_self">Th28</a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E67:60"/>
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E68:60"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p3">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E69:60"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p5">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T26" target="_self">Th26</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E70:60"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="sprect_3.html#E13:60"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T27" target="_self">Th27</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E71:60"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:60"><i><font color="Green">E38</font></i></a>, , , , , , <a class="ref" href="sprect_3.html#T19" target="_self">Th19</a>, , , , <a class="ref" href="jordan5c.html#T28">JORDAN5C:28</a></i>;<br/>
<a NAME="E72:60"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p2">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="sprect_3.html#T28" target="_self">Th28</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E73:60"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>


</div>
