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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:30"/><i><font color="Green">E47</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">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> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K3" target="_self">North_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:30"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K4" target="_self">South_Arc</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E6:30"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8">Lim_inf</a> <span class="p1">(<span class="default"><a href="jordan19.html#K1" target="_self">Upper_Appr</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:30"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8">Lim_inf</a> <span class="p1">(<span class="default"><a href="jordan19.html#K2" target="_self">Lower_Appr</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>;<br/>
<a NAME="E8:30"/>
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> 0 )
 
;<br/>
<a NAME="E9:30"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E10:30"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>reconsider </b><font color="Maroon">rr</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:30_1"/><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 ;<br/><a NAME="E3:30_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then reconsider </b><font color="Maroon">G</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="goboard6.html#T5">GOBOARD6:5</a></i>;<br/><b>consider </b><font color="Maroon">k1</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="E6:30_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K1" target="_self">Upper_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#D11">KURATO_2:def 11</a></i>;<br/><b>consider </b><font color="Maroon">k2</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="E8:30_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K2" target="_self">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#D11">KURATO_2:def 11</a></i>;<br/><b>set </b><font color="Maroon">k</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>;<br/><a NAME="E9:30_1"/><i><font color="Green">E72</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>set </b><font color="Maroon">z'</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">z</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>;<br/><a NAME="E10:30_1"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E11:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/><a NAME="E12:30_1"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,1
 <b>by </b><i><a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E13:30_1"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="power.html#T59">POWER:59</a></i>;<br/><b>then reconsider </b><font color="Maroon">m'</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/><a NAME="E15:30_1"/><i><font color="Green">E73</font></i>: 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>set </b><font color="Maroon">N</font> = 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E16:30_1"/>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/><a NAME="E17:30_1"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E18:30_1"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E19:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E20:30_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E21:30_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E22:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E23:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E24:30_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E25:30_1"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E26:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E27:30_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E28:30_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E29:30_1"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E30:30_1"/><b>then </b>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E31:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E32:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan19.html#K1" target="_self">Upper_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E33:30_1"/><b>then </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E35:30_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E36:30_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">p1</font> = <font color="Maroon">p1</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>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E39:30_1"/><i><font color="Green">E79</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T63">JORDAN1G:63</a></i>;<br/><b>then consider </b><font color="Maroon">i1</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="E41:30_1"/><i><font color="Green">E80</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E42:30_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E45:30_1"/><i><font color="Green">E82</font></i>: 
 <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T4">JORDAN1G:4</a></i>;<br/><a NAME="E46:30_1"/>
<font color="Maroon">i1</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="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E47:30_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E48:30_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">ii1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jj1</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="E50:30_1"/><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii1</font>,<font color="Maroon">jj1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E51:30_1"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj1</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E52:30_1"/>
(  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 &amp;  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 )
 <b>by </b><i><a class="ref" href="topreal5.html#T22">TOPREAL5:22</a>, <a class="ref" href="topreal5.html#T23">TOPREAL5:23</a></i>;<br/><a NAME="E53:30_1"/><b>then </b><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E54:30_1"/><i><font color="Green">E98</font></i>: 
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E55:30_1"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m'</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E56:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</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="E57:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m'</font> <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="E58:30_1"/><b>then </b>
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E59:30_1"/><b>then </b><i><font color="Green">E99</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/><a NAME="E60:30_1"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E61:30_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E62:30_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E63:30_1"/><b>then </b><i><font color="Green">E100</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E64:30_1"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, , </i>;<br/><a NAME="E65:30_1"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E66:30_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E67:30_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E68:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T53">JGRAPH_1:53</a></i>;<br/><a NAME="E69:30_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E70:30_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E71:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E72:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E73:30_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 ;<br/><a NAME="E74:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/><a NAME="E75:30_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E76:30_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E77:30_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E78:30_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan19.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan19.html#E2:30"><i><font color="Green">E47</font></i></a>, <a class="ref" href="sppol_2.html#T48">SPPOL_2:48</a></i>;<br/><a NAME="E79:30_1"/><i><font color="Green">E109</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E80:30_1"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E81:30_1"/><b>then </b>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E82:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E83:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" target="_self">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E84:30_1"/><b>then </b>
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E86:30_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E87:30_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">p2</font> = <font color="Maroon">p2</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>reconsider </b><font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E90:30_1"/><i><font color="Green">E112</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T64">JORDAN1G:64</a></i>;<br/><b>then consider </b><font color="Maroon">i2</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="E92:30_1"/><i><font color="Green">E113</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E93:30_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>reconsider </b><font color="Maroon">d2</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E96:30_1"/><i><font color="Green">E115</font></i>: 
 <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T5">JORDAN1G:5</a></i>;<br/><a NAME="E97:30_1"/>
<font color="Maroon">i2</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="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E98:30_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E99:30_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">ii2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jj2</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="E101:30_1"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii2</font>,<font color="Maroon">jj2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E102:30_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj2</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E103:30_1"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, , </i>;<br/><a NAME="E104:30_1"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E105:30_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E106:30_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E107:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="txt" href="jordan19.html#E10:30"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_1.html#T53">JGRAPH_1:53</a></i>;<br/><a NAME="E108:30_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="txt" href="jordan19.html#E8:30_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E109:30_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E110:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E111:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E9:30_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E112:30_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 ;<br/><a NAME="E113:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/><a NAME="E114:30_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="txt" href="jordan19.html#E15:30_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E115:30_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E116:30_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E117:30_1"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E2:30_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sppol_2.html#T48">SPPOL_2:48</a></i>;<br/><b>set </b><font color="Maroon">Gij</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font>;<br/><b>set </b><font color="Maroon">Gji</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font>;<br/><b>reconsider </b><font color="Maroon">Gij'</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font>, <font color="Maroon">Gji'</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E119:30_1"/><i><font color="Green">E126</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii1</font> &amp; <font color="Maroon">ii1</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="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font> &amp; <font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan19.html#E3:30"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E120:30_1"/><i><font color="Green">E127</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii2</font> &amp; <font color="Maroon">ii2</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="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj2</font> &amp; <font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan19.html#E3:30_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E121:30_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 )
 <b>by </b><i><a class="ref" href="jordan1e.html#T19">JORDAN1E:19</a></i>;<br/><a NAME="E122:30_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E123:30_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E124:30_1"/><b>then </b><i><font color="Green">E128</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E125:30_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T225">XREAL_1:225</a></i>;<br/><a NAME="E126:30_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E127:30_1"/><i><font color="Green">E131</font></i>: 
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><div><i><font color="Green">E132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1"/>
1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ii1</font>
 ;<br/><a NAME="E2:30_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">ii1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E5:30"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan19.html#E42:30_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:30_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:30_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E48:30_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:30_1_1"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E7:30_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E8:30_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:30_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan19.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>;<br/><a NAME="E10:30_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E11:30_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:30_1_1"/><b>then </b>
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E13:30_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2"/>
<font color="Maroon">ii1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:30_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">ii1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> = 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan19.html#D2" target="_self">Def2</a>, <a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan19.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan1j.html#T46">JORDAN1J:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</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="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T7">JORDAN1F:7</a></i>
;<br/><a NAME="E5:30_1_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E2:30"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan19.html#E41:30_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><b>hence </b><a NAME="E6:30_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_3"/>
<font color="Maroon">ii2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:30_1_3"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">ii2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> = 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E35:30_1"><i><font color="Green">E77</font></i></a>, , <a class="ref" href="jordan1j.html#T47">JORDAN1J:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</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="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T8">JORDAN1F:8</a></i>
;<br/><a NAME="E5:30_1_3"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E2:30_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan19.html#E41:30_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><b>hence </b><a NAME="E6:30_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_4"/>
<font color="Maroon">ii2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:30_1_4"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">ii2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_1_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E27:30_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan19.html#E42:30_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:30_1_4"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:30_1_4"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E48:30_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:30_1_4"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E7:30_1_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E8:30_1_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E9:30_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><a NAME="E11:30_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E12:30_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:30_1_4"/><b>then </b>
 <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E14:30_1_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E132:30_1"/><i><font color="Green">E145</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E42:30_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/><i><font color="Green">E146</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><i><font color="Green">E147</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E148</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><i><font color="Green">E149</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E137:30_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E138:30_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E139:30_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 )
 <b>by </b><i><a class="txt" href="jordan19.html#E4:30"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan19.html#E24:30_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E140:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E141:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><a NAME="E142:30_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E45:30_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E143:30_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E144:30_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E145:30_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 )
 <b>by </b><i><a class="txt" href="jordan19.html#E4:30"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan19.html#E24:30_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E146:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E147:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><a NAME="E148:30_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E45:30_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E149:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T49">JGRAPH_1:49</a></i>;<br/><a NAME="E150:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E151:30_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">Gij'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E152:30_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E153:30_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T49">JGRAPH_1:49</a></i>;<br/><a NAME="E154:30_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E155:30_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">Gji'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E156:30_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E157:30_1"/><i><font color="Green">E154</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E28:30_1"><i><font color="Green">E76</font></i></a>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E158:30_1"/><i><font color="Green">E155</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E6:30"><i><font color="Green">E59</font></i></a>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E159:30_1"/><i><font color="Green">E156</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E28:30_1"><i><font color="Green">E76</font></i></a>, , <a class="txt" href="jordan19.html#E51:30_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E160:30_1"/><i><font color="Green">E157</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E6:30"><i><font color="Green">E59</font></i></a>, , <a class="txt" href="jordan19.html#E51:30_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_9_1"/>
( <font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font> or <font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">jj2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_1"><b>suppose </b></a><a NAME="E1:30_1_9_1_1"/>
<font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font>
 ;<br/><div class="add"><a NAME="E2:30_1_9_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan19.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E35:30_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan19.html#E50:30_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="jordan19.html#T3" target="_self">Th3</a>, , , <a class="ref" href="jordan15.html#T50">JORDAN15:50</a></i>;<br/><a NAME="E3:30_1_9_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_9_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_1_9_1_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_9_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_9_1_1_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> or <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan19.html#E69:30_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_9_1_1_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_9_1_1_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E53:30_1"><i><font color="Green">E97</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_9_1_1_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_9_1_1_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E54:30_1"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_9_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_9_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E67:30_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_2"><b>suppose </b></a><a NAME="E1:30_1_9_1_2"/>
<font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">jj2</font>
 ;<br/><div class="add"><a NAME="E2:30_1_9_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="jordan19.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan19.html#E6:30_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan19.html#E35:30_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan19.html#E36:30_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan19.html#E39:30_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan19.html#E50:30_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="jordan19.html#T3" target="_self">Th3</a>, , , </i>;<br/><a NAME="E3:30_1_9_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_9_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_1_9_1_2_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_9_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_9_1_2_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> or <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan19.html#E69:30_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_9_1_2_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_9_1_2_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E63:30_1"><i><font color="Green">E100</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_9_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_9_1_2_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_9_1_2_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E59:30_1"><i><font color="Green">E99</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_9_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_9_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E67:30_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E162:30_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E12:30"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard6.html#T96">GOBOARD6:96</a></i>;<br/>
<a NAME="E13:30"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>reconsider </b><font color="Maroon">rr</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>assume </b><a NAME="E2:30_2"/><i><font color="Green">E164</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> )
 ;<br/><a NAME="E3:30_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then reconsider </b><font color="Maroon">G</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="goboard6.html#T5">GOBOARD6:5</a></i>;<br/><b>consider </b><font color="Maroon">k1</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="E6:30_2"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K1" target="_self">Upper_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#D11">KURATO_2:def 11</a></i>;<br/><b>consider </b><font color="Maroon">k2</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="E8:30_2"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> holds <br/><span class="p1">(<span class="default"><a href="jordan19.html#K2" target="_self">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#D11">KURATO_2:def 11</a></i>;<br/><b>set </b><font color="Maroon">k</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>;<br/><a NAME="E9:30_2"/><i><font color="Green">E168</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><b>set </b><font color="Maroon">z'</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">z</font> =  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>;<br/><a NAME="E10:30_2"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E11:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="real_2.html#T143">REAL_2:143</a></i>;<br/><a NAME="E12:30_2"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="power.html#K6">log</a> 2,1
 <b>by </b><i><a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E13:30_2"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="power.html#T59">POWER:59</a></i>;<br/><b>then reconsider </b><font color="Maroon">m'</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K5">log</a> 2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p5">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/><a NAME="E15:30_2"/><i><font color="Green">E169</font></i>: 
2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>set </b><font color="Maroon">N</font> = 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E16:30_2"/>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/><a NAME="E17:30_2"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a></i>;<br/><a NAME="E18:30_2"/><b>then </b>
 <a href="power.html#K5">log</a> 2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> 2,<span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T63">POWER:63</a></i>;<br/><a NAME="E19:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_ff.html#T12">PRE_FF:12</a></i>;<br/><a NAME="E20:30_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E21:30_2"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E22:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E23:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E24:30_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E25:30_2"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E26:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E27:30_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E28:30_2"/><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E29:30_2"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E30:30_2"/><b>then </b>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="jordan19.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E31:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E32:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan19.html#K1" target="_self">Upper_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E33:30_2"/><b>then </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E35:30_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E36:30_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">p1</font> = <font color="Maroon">p1</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>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>set </b><font color="Maroon">f</font> =  <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E39:30_2"/><i><font color="Green">E175</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T63">JORDAN1G:63</a></i>;<br/><b>then consider </b><font color="Maroon">i1</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="E41:30_2"/><i><font color="Green">E176</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E42:30_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E45:30_2"/><i><font color="Green">E178</font></i>: 
 <a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T4">JORDAN1G:4</a></i>;<br/><a NAME="E46:30_2"/>
<font color="Maroon">i1</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="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E47:30_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E48:30_2"/><b>then </b><i><font color="Green">E179</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">ii1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jj1</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="E50:30_2"/><i><font color="Green">E180</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii1</font>,<font color="Maroon">jj1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E51:30_2"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj1</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E2:30"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E52:30_2"/>
(  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 &amp;  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 )
 <b>by </b><i><a class="ref" href="topreal5.html#T22">TOPREAL5:22</a>, <a class="ref" href="topreal5.html#T23">TOPREAL5:23</a></i>;<br/><a NAME="E53:30_2"/><b>then </b><i><font color="Green">E182</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E54:30_2"/><i><font color="Green">E183</font></i>: 
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E55:30_2"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m'</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E56:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</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="E57:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m'</font> <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="E58:30_2"/><b>then </b>
2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/><a NAME="E59:30_2"/><b>then </b><i><font color="Green">E184</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/><a NAME="E60:30_2"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E61:30_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E62:30_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E63:30_2"/><b>then </b><i><font color="Green">E185</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E64:30_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E2:30"><i><font color="Green">E47</font></i></a>, </i>;<br/><a NAME="E65:30_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E66:30_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E67:30_2"/><b>then </b><i><font color="Green">E186</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E68:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T53">JGRAPH_1:53</a></i>;<br/><a NAME="E69:30_2"/><b>then </b><i><font color="Green">E187</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="jordan19.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E70:30_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="jordan19.html#D2" target="_self">Def2</a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E71:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E72:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E73:30_2"/><b>then </b><i><font color="Green">E188</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 ;<br/><a NAME="E74:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/><a NAME="E75:30_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="txt" href="jordan19.html#E4:30"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E76:30_2"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E77:30_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E78:30_2"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="jordan19.html#E3:30"><i><font color="Green">E48</font></i></a>, <a class="ref" href="sppol_2.html#T48">SPPOL_2:48</a></i>;<br/><a NAME="E79:30_2"/><i><font color="Green">E193</font></i>: 
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E80:30_2"/>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E81:30_2"/><b>then </b>
 <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E82:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E83:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan19.html#K2" target="_self">Lower_Appr</a> <font color="Maroon">C</font></span>)</span> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E84:30_2"/><b>then </b>
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E86:30_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E87:30_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</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">p2</font> = <font color="Maroon">p2</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>reconsider </b><font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E90:30_2"/><i><font color="Green">E196</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T64">JORDAN1G:64</a></i>;<br/><b>then consider </b><font color="Maroon">i2</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="E92:30_2"/><i><font color="Green">E197</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E93:30_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>reconsider </b><font color="Maroon">d2</font> = <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E96:30_2"/><i><font color="Green">E199</font></i>: 
 <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan1g.html#T5">JORDAN1G:5</a></i>;<br/><a NAME="E97:30_2"/>
<font color="Maroon">i2</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="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E10:30"><i><font color="Green">E66</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E98:30_2"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E10:30"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E99:30_2"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">ii2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">jj2</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="E101:30_2"/><i><font color="Green">E201</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ii2</font>,<font color="Maroon">jj2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E102:30_2"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj2</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E103:30_2"/>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan19.html#E10:30"><i><font color="Green">E66</font></i></a>, , </i>;<br/><a NAME="E104:30_2"/><b>then </b>
 <a href="topreal6.html#K1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E105:30_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">d1</font>,<font color="Maroon">d2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><a NAME="E106:30_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E107:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T53">JGRAPH_1:53</a></i>;<br/><a NAME="E108:30_2"/><b>then </b><i><font color="Green">E204</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E109:30_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p2'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E110:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 8
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E111:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 4</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E112:30_2"/><b>then </b><i><font color="Green">E205</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 ;<br/><a NAME="E113:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/><a NAME="E114:30_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E115:30_2"/><b>then </b><i><font color="Green">E207</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E116:30_2"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E117:30_2"/><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="sppol_2.html#T48">SPPOL_2:48</a></i>;<br/><b>set </b><font color="Maroon">Gij</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font>;<br/><b>set </b><font color="Maroon">Gji</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font>;<br/><b>reconsider </b><font color="Maroon">Gij'</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font>, <font color="Maroon">Gji'</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E119:30_2"/><i><font color="Green">E210</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii1</font> &amp; <font color="Maroon">ii1</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="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font> &amp; <font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan19.html#E6"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E120:30_2"/><i><font color="Green">E211</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ii2</font> &amp; <font color="Maroon">ii2</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="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj2</font> &amp; <font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E121:30_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 )
 <b>by </b><i><a class="ref" href="jordan1e.html#T19">JORDAN1E:19</a></i>;<br/><a NAME="E122:30_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E123:30_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E124:30_2"/><b>then </b><i><font color="Green">E212</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E125:30_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T225">XREAL_1:225</a></i>;<br/><a NAME="E126:30_2"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E127:30_2"/><i><font color="Green">E215</font></i>: 
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><div><i><font color="Green">E216</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_1"/>
1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ii1</font>
 ;<br/><a NAME="E2:30_2_1"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">ii1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_2_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan19.html#E6:30"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:30_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:30_2_1"/><b>then </b><i><font color="Green">E218</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:30_2_1"/><i><font color="Green">E219</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E7:30_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E8:30_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:30_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, , , , <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>;<br/><a NAME="E10:30_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E11:30_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:30_2_1"/><b>then </b>
 <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E13:30_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><i><font color="Green">E220</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_2"/>
<font color="Maroon">ii1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:30_2_2"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">ii1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> = 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan19.html#E5"><i><font color="Green">Lemma44</font></i></a>, , , , <a class="ref" href="jordan1j.html#T46">JORDAN1J:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</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="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T7">JORDAN1F:7</a></i>
;<br/><a NAME="E5:30_2_2"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan19.html#E3:30"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><b>hence </b><a NAME="E6:30_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E252</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_3"/>
<font color="Maroon">ii2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E2:30_2_3"/><b>then </b><i><font color="Green">E253</font></i>: 
<font color="Maroon">ii2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan1a.html#T94">JORDAN1A:94</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> = 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan19.html#E7:30"><i><font color="Green">E65</font></i></a>, , , , <a class="ref" href="jordan1j.html#T47">JORDAN1J:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</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="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1f.html#T8">JORDAN1F:8</a></i>
;<br/><a NAME="E5:30_2_3"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><b>hence </b><a NAME="E6:30_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan19.html#E10:30"><i><font color="Green">E66</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E254</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_4"/>
<font color="Maroon">ii2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:30_2_4"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">ii2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:30_2_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:30_2_4"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:30_2_4"/><b>then </b><i><font color="Green">E256</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:30_2_4"/><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E7:30_2_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E8:30_2_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E9:30_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i>, , , , <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>;<br/><a NAME="E11:30_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E12:30_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:30_2_4"/><b>then </b>
 <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E14:30_2_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E132:30_2"/><i><font color="Green">E258</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">rr</font> <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/><i><font color="Green">E259</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><i><font color="Green">E260</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><i><font color="Green">E261</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/><i><font color="Green">E262</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E137:30_2"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E138:30_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E139:30_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 )
 <b>by </b><i><a class="txt" href="jordan19.html#E5:30"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E140:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E141:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><a NAME="E142:30_2"/><b>then </b><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E143:30_2"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T51">JGRAPH_1:51</a></i>;<br/><a NAME="E144:30_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 <b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/><a NAME="E145:30_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 4 )
 <b>by </b><i><a class="txt" href="jordan19.html#E5:30"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E146:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E147:30_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><a NAME="E148:30_2"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E149:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T49">JGRAPH_1:49</a></i>;<br/><a NAME="E150:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E151:30_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">Gij'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E152:30_2"/><b>then </b><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E153:30_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T49">JGRAPH_1:49</a></i>;<br/><a NAME="E154:30_2"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E155:30_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">Gji'</font>,<font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="sppol_1.html#T62">SPPOL_1:62</a></i>;<br/><a NAME="E156:30_2"/><b>then </b><i><font color="Green">E266</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E157:30_2"/><i><font color="Green">E267</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E158:30_2"/><i><font color="Green">E268</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="ref" href="jordan19.html#T2" target="_self">Th2</a>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E159:30_2"/><i><font color="Green">E269</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><a NAME="E160:30_2"/><i><font color="Green">E270</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">rr</font>
 <b>by </b><i><a class="ref" href="jordan19.html#T2" target="_self">Th2</a>, , , <a class="ref" href="topreal3.html#T28">TOPREAL3:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_9_1"/>
( <font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font> or <font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">jj2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_1"><b>suppose </b></a><a NAME="E1:30_2_9_1_1"/>
<font color="Maroon">jj2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">jj1</font>
 ;<br/><div class="add"><a NAME="E2:30_2_9_1_1"/><b>then </b><i><font color="Green">E271</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , <a class="ref" href="jordan19.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="jordan15.html#T51">JORDAN15:51</a></i>;<br/><a NAME="E3:30_2_9_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_9_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_2_9_1_1_1"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_9_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_9_1_1_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span> or <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_2_9_1_1_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_2_9_1_1_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_2_9_1_1_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii2</font>,<font color="Maroon">jj1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_2_9_1_1_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_2_9_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:30_2_9_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_2"><b>suppose </b></a><a NAME="E1:30_2_9_1_2"/>
<font color="Maroon">jj1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">jj2</font>
 ;<br/><div class="add"><a NAME="E2:30_2_9_1_2"/><b>then </b><i><font color="Green">E274</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , <a class="ref" href="jordan19.html#T3" target="_self">Th3</a>, , , , </i>;<br/><a NAME="E3:30_2_9_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_9_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_2_9_1_2_1"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p0">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_9_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_9_1_2_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span> or <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_2_9_1_2_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_2_9_1_2_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_9_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_2_9_1_2_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ii1</font>,<font color="Maroon">jj2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">C</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p5">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span>,<font color="Maroon">m'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_2_9_1_2_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_2_9_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:30_2_9_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E162:30_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E15:30"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard6.html#T96">GOBOARD6:96</a></i>;<br/>
<a NAME="E16:30"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E17:30"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:30"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E19:30"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E20:30"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>


</div>
